Fete of Combinatorics and Computer Science

BOLYAI SOCIETY MATHEMATICAL STUDIES

BOLYAI socrsrv MATHEMATICAL STUDIES Series Editor: GaborFejes T6th Publication Board: Gyula O.H. Katona- Laszlo Lovasz . PeterPalPalfy AndrasRecski . AndrasStipsicz . Domokos Szasz Managing Editor: DezsO Mikl6s 1. Combinatorics, Paul Erdlls is Eighty,Vol. 1 D. Miklos, V.T. S6s,T.Szonyi(Eds.) 2. Combinatorics, Paul Erdlls is Eighty,Vol. 2 D. Mikl6s,V.T. S6s,T.Szonyi(Eds.) 3. Extremal Problems for Finite Sets P. Frankl, Z. Fiiredi,G. Katona, D. Mikl6s(Eds.) 4. Topologywith Applications A.Csaszar(Ed.) 5. Approximation Theory and Function Series P.Vertesi, L. Leindler, Sz. Revesz, J. Szabados, V. Totik(Eds.) 6. Intuitive Geometry I. Barany, K. Boroczky (Eds.) 7. Graph Theory and Combinatorial Biology L. Lovasz, A. Gyarfas, G. Katona, A. Recski (Eds.) 8. Low Dimensional Topology K. Boroczky, [r.,W Neumann, A.Stipsicz (Eds.) 9. Random Walks P. Revesz, B. T6th (Eds.) 10. Contemporary Combinatorics B. Bollobas (Ed.) 11. Paul Erdlls and His Mathematics 1+11 G. Halasz, L. Lovasz, M. Simonovits, V. T. S6s (Eds.) 12. Higher Dimensional Varietiesand Rational Points K. Boroczky, Ir., J. Kollar, T. Szarnuely (Eds.) 13. Surgery on Contact 3-Manifolds and Stein Surfaces B.Ozbagci, A. I. Stipsicz 14. A Panorama of Hungarian Mathematics in the Twentieth Century,Vol.1 J. Horvath (Ed.) 15. More Sets, Graphs and Numbers E. Gy6ri,G. Katona,L. Lovasz (Eds.) 16. Entropy,Search, Complexity I. Csiszar, G. Katona,G. Tardos(Eds.) 17. Horizons of Combinatorics E. Gy6ri,G. Katona,L. Lovasz (Eds.) 18. Handbook of Large-Scale Random Networks B.Bollobas, R. Kozma, D. Miklos (Eds.) 19. Building Bridges M. Grotschel,G. Katona (Eds.)

Gyula o. H. Katona Alexander Schrijver Tamas Szonyi (Eds.)

Fete of Combinatorics and Computer Science

~ Springer

JANOS BOLYAI MATHEMATICAL SOCIETY

GyulaO. H. Katona

Managing Editor:

AlfredRenyi Institute of Mathematics HungarianAcademy of Sciences Realtanoda u. 13-15 Budapest1053 Hungary e-mail: [email protected]

Gabor Sagi

AlexanderSchrijver

AlfredRenyi Instituteof Mathematics HungarianAcademy of Sciences Realtanoda u. 13-15 Budapest 1053 Hungary e-mail: [email protected]

Centre for Mathematics and Computer Science Science Park 123 1098 XGAmsterdam The Netherlands e-mail: [email protected]

Tamas Szonyi Eotvos University Department of Computer Science PazmanyPetersetany l/C Budapest1117 Hungary e-mail: [email protected]

Mathematics Subject Classification (2000): 05XX, 68XX

Libraryof CongressControl Number:2010928477

ISSN 1217-4696 ISBN 978-3-642-13579-8 SpringerBerlinHeidelberg NewYork ISBN 978-963-9453-12-8 JanosBolyai Mathematical Society, Budapest This work is subject to copyright.All rights are reserved,whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilmor in any other way, and storagein data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Lawof September 9,1965,in its current version, and permission for use must always be obtainedfrom Springer. Violations are liablefor prosecutionunder the German CopyrightLaw. Springeris a part of SpringerScience-sBusiness Media springer.com © 2010JanosBolyai Mathematical Societyand Springer-Verlag Printed in Hungary The use of general descriptive names, registered names, trademarks,etc. in this publicationdoes not imply. even in the absenceof a specific statement,that such namesare exemptfrom the relevant protectivelawsand regulations and thereforefree for generaluse. Cover design:WMXDesign GmbH,Heidelberg Printed on acid-freepaper 44/3142/db- 5432 10

CONTENTS

CONTENTS . ...... . ... . .. . . . . . . . .... .. . . . . . . .. . . .. .. . .. . .. . .. .. .... .. PREFACE. . . . ... ... ....... . . . ... .... . . . ... . ..... . .. .. . .. . ... . .. .. . . .. N. ALON and N. WORMALD: High Degree Graphs Contain LargeStar Factors S. BUTLER and R. GRAHAM: Iterated Triangle Partitions F. CHUNG and W . ZHAO: PageRank and Random Walks on Graphs . J . CIBULKA, J . KYNCL, V . MESZAROS , R. STOLAR and P. VALTR: Solution of Peter Winkler's Pizza Problem B . CSABA, I. LEVITT, J. NAGy-GYORGY and E . SZEMEREDI: Tight Bounds for Embedding Bounded Degree Trees G . ELEK: Betti Numbers are Testable T . JORDAN : Rigid and Globally Rigid Graphs with Pinned Vertices .. G. KALAl: Noise Sensitivity and Chaos in Social Choice Theory ..... A. KOSTOCHKA , M. KUMBHAT and V . RODL: Coloring Uniform Hypergraphs with Small Edge Degrees E. MARCHANT and A . THOMASON: Extremal Graphs and Multigraphs with Two Weighted Colours V. RODL and M . SCHACHT: Regularity Lemmas for Graphs B . SZEGEDY: Edge Coloring Models as Singular Vertex Coloring Models T .-L. WONG , D. YANG and X . ZHU: List Total Weighting of Graphs OPEN PROBLEMS

5 7 9 23 43 63 95 139 151 173 213 239 287 327 337 355

PREFACE

This book is the second volume of mathematical papers to commemorate the 60th birthday of Laszlo ('Laci ') Lovasz on March 9, 2008. Prominent mathematicians that were inspired by Laci were invited to contribute to the celebrations of this milestone in the Summer of 2008. The response was overwhelming, and therefore two conferences were organized , the first from 5-9 August 2008 in Budapest, and the second the week after, from 11-15 August 2008 in Keszthely, at the borders of Lake Balaton. The first meeting was baptized Building Bridges - Between Mathematics and Computer Sci ence, the second Fete of Combinatorics and Computer Science. These names also adorn the two volumes published to commemorate the celebrations. The two names very well reflect the power and the beauty of Laci's work. Many of his results indeed can be characterized by building new, powerful bridges between and within mathematics and computer science, and these are really feasts to the combinatorial and algorithmic mind. Several of the methods found by Lovasz have opened up new areas of research in discrete mathematics and algorithmics. Often , these methods were obtained by Lovasz by developing beautiful new techniques based on algebra , geometry, or topology, thus laying fundaments for the important role of classical mathematics in more modern fields like combinatorics, optimization, algorithmics, and complexity. Laci Lovasz has obtained so many pioneering results that it would be quite impossible to describe a reasonable part of it . Let us just give a brief selection. He solved several important open problems in discrete mathematics, like the perfect graph conjecture, Kneser's conjecture, the Shannon capacity problem , the matroid matching problem, and the matching lattice problem. He proved the now called 'Lovasz Local Lemma' , basic in probabilistic combinatorics and randomized algorithms. He pointed out the significance of the ellipsoid method and semidefinite programming for combinatorial optimization, and he designed the lattice basis reduction method, with a wealth of applications in discrete mathematics, combinatorial optimization , integer programming, number theory, algebra, and cryptography. Other basic algorithmic results were found for volume computation and

8

Preface

communication complexity. Recently, Laci was the main inspirator of the new area of graph limits, with its connections to extremal combinatorics and mathematical physics. Laci's deep insight in the combinatorial and algorithmic potentials of geometry and topology is unique. Several of his results are ground-breaking, and there seems little chance that they would have been detected without him. They have inspired and directed the work of a huge part of the researchers in discrete mathematics, optimization, and algorithmics . Laci's reputation is beyond any measure not only because of his results and insights , but also because of his very pleasant modesty, his willingness to cooperate and to exchange ideas, and the extreme transparency of his lectures, which form usually exciting eye-openers. The inspiration and charismatic ambiance offered by Laci and his wife Kati have given them many friends and collaborators, all over the world and in a wide range in mathematics and computer science. We are very happy that many of them have reacted enthusiastically to our invitation to contribute to the volumes, and that we received so many excellent articles. Like the first volume, also this second volume forms an expression of the impact of Laci Lovasz's work and of the inspiration given by him to many scientists world-wide. Budapest August 2009

Gyula Katona Alexander Schrijver Tamas Szonyi

Fete of Combinatorics, pp. 9-21.

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

HIGH DEGREE GRAPHS CONTAIN LARGE-STAR FACTORS

NOGA ALON* and NICHOLAS WORMALDt Dedicated to Laszl6 Loudsz, for his 60th birthday

We show that any finite simple graph with minimum degree d contains a spanning star forest in which every connected component is of size at least n((dj logd)l /3). This settles a problem of Havet, Klazar, Kratochvil, Kratsch and Liedloff.

Dedication This paper is dedicated to Lad Lovasz, for his 60th birthday. It settles a problem presented by Jan Kratochvil at the open problems session of the meeting Building Bridges, which took place in Budapest in August 2008, celebrating this birthday. The Lovasz Local Lemma is applied extensively throughout the proof. This work is therefore a typical example illustrating the immense influence of Laci, who not only provided the community with powerful tools and techniques, but also stimul ated research by his books, lectures and organization of conferences.

1. INTRODUCTION All graphs considered here are finite and simple. A star is a tree with one vertex, the center, adjacent to all the others, which are leaves. A star factor of a graph G is a spanning forest of G in which every connected component is a star. It is easy to see that any graph with positive minimum degree contains a star factor in which every component is a star with at least one "Research supported in part by ERC advanced grant, by a USA-Israel BSF grant, by NSF grant CCF 0832797 and by the Ambrose Monell Foundation. tSupported by the Canada Research Chairs Program and NSERC .

10

N. Alon and N. Worm ald

edge. A conjecture of Havet et al. [9], communicated to us by Jan Kratochvil [10], asserts that if the minimum degree is large then one can ensure that all stars are large. More precisely, they conjectured that there is a function g(d) that tends to infinity as d tends to infinity, so that every graph with minimum degree d contains a star factor in which every star contains at least g(d) edges. Our main result shows that this is indeed the case, for a function g(d) that grows moderately quickly with d, as follows. Theorem 1.1. There exists an absolute positive constant c so that for all d ~ 2, every graph with minimum degree d contains a star factor in which every star has at least cd1/ 3 j (log d) 1/3 edges.

The motivation for the conjecture of Havet et al. arises in the running time analysis of a recent exact exponential time algorithm for the so called L(2 , I)-labeling problem of graphs. See [9] for more details. As preparation for the proof of the main result, we prove the following simpler statement. Theorem 1.2. There exists an absolute positive constant c' such that for all d ~ 2, every d-regular graph contains a star factor in which every star has at least c'd] log d edges. This is optimal, up to the value of the constant c'.

Throughout the paper we make no attempt to optimize the absolute constants. To simplify the presentation we omit all floor and ceiling signs whenever these are not crucial. We may and will assume, whenever this is needed, that the minimum degree d considered is sufficiently large. It is easy to find, in any graph with all vertices of degree at least 1, a st ar factor with st ars of at least two vertices each, so the theorems then follow for all d ~ 2. All logarithms are in the natural base, unless otherwise specified. Our notation is standard. In particular, for a graph G = (V, E) and a vertex v E V, we let Nc(v) denote the set of all neighbors of v in the graph G, and let dc(v) = INc(v)1 denote the degree of v in G. For X c V, Nc(X) = UxEXNC(X) is the set of all neighbors of the members of X . The rest of this short paper is organized as follows. In Section 2 we present the simple proof of Theorem 1.2, and in Section 3 the proof of the main result. Section 4 contains some concluding remarks and open problems.

11

High Degree Graphs Contain Large-Star Factors

2.

REGULAR GRAPHS

Proof of Theorem 1.2 . Let G = (V, E) be a d-regular graph. Put p = (2 + 210gd)jd and let C be a random set of vertices obtained by picking each vertex of G, randomly and independently, to be a member of C, with probability p. We will show that, with positive probability, some such set C will be a suitable choice for the set of centres of the stars in the desired star factor . For each vertex v E V, let Av be the event that either v has no neighbors in C or v has more than 3pd neighbors in C. By the standard known estimates for binomial distributions (c.f., e.g., [4] , Theorem A.1.12), the probability of each event A v is at most (1 - p)d + (e2j27)2+2 log d < e-P + (lje)2+2logd < Ijed 2. Moreover, each event A v is mutually independent of all events Au except those that satisfy NG(v) n NG(u) =1= 0. As there are at most d(d - 1) < d2 such vertices u we can apply the Lovasz Local Lemma (c.f., e.g., [4], Corollary 5.1.2) to conclude that with positive probability none of the events Av holds. Therefore, there is a choice of a set C C V so that for every vertex v, 0 < N G ( v) n C 3pd = 6 + 6 log d.

I

I : :;

Fix such a set C, and let B be the bipartite graph whose two classes of vertices are C and V \ C, where each v E C is adjacent in B to all vertices u E V \ C which are its neighbors in G. By the choice of C, for every vertex v E C, dB(V) ~ d - 6 - 610gd and for every vertex u E V \ c, o < dB(U) :::; 6 + 6 log d. It thus follows by Hall's theorem that one can assign to each vertex v E C a set consisting of

d - 6 - 610gd 6 + 6 log d

d 7 log d

-,-------:-:--~>--

of its neighbors in V\ C, where no member of V\ C is assigned to more than one such v. (To see this from the standard version of Hall's theorem, split each vertex v in C into (d - 6 - 6 log d)j (6 + 6 log d) identical 'sub-vertices', each with the same neighbours in V \ C as v, and find a matching that hits every sub-vertex using Hall's theorem . Then for each v E C, coalesce the subvertices of v back together to form v.) By assigning each unassigned vertex U of V \ C arbitrarily to one of its neighbors in C (note that there always is such a neighbor) we get the required star factor, in which the centers are precisely the members of C and each star contains more than dj(710gd) edges.

12

N. Alon and N. Wormald

It remains to show that the above estimate is optimal, up to a constant factor. Note that the centers of any star factor form a dominating set in the graph, and thus if the minimum size of a dominating set in a d-regular graph on n vertices is at least O(nlogdld), then the star factor must contain a component of size at most O(dl log d). It is not difficult to check that the minimum size of a dominating set in a random d-regular graph on n vertices is 8(n log did) with high probability. In fact, if c < 1, the expected number of dominating sets of size k = n ( c + o(1)) (log d)I d tends to 0 for d sufficiently large . We give some details of verifying this claim. One can use the standard pairing or configuration model of random d-regular graphs, in which there are n buckets with d points in each bucket. It is enough to prove the result for the multigraph arising from taking a random pairing of the points and regarding the buckets as vertices (see e.g. [12] for details) . The expected number of dominating sets S of size k with m edges from S to N(S) is '=

A.

(n) ( ) k f k, m

(mrr- l (kd _ .)) M(kd - m)M( (n - k)d - m) M (nd) i=O

1,

where f(k , m) is the coefficient of x m in ((1 + x)d -1) n-k and M(r) is the number of pairings or perfect matchings of an even number r of points, i.e. (r - 1) (r - 3) . .. 1. In the above formula, the binomial chooses the k buckets of S, f(k, m) is the number of ways to choose m points in the other n - k buckets such that at least one point comes from each bucket (so that S dominates the graph) , and the next factor counts the ways to pair those points with points in buckets in S . The other factors in the numerator count the ways to pair up the remaining points, first within S, and then within the rest of the graph. The denominator is the total number of pairings. We may use standard methods to see that f(k, m) ::; ((1 + x)d -1) n-k x - m for all real x > 0 (since I's coefficients are all nonnegative). We set x = kin and take k = nc(log d)I d, which is justified by regarding c as a function of n that tends to a limit equal to the value c referred to in the claim above. This gives

A < (n) ((1 kl)d _ 1) n-k (~)m (kd)! + n k (kd - m)! - k M(kd - m)M( (n - k)d - m) M(nd)

13

High Degree Graphs Contain Large-Star Factors

Considering replacing m by m + 2 shows this expression is maximised when n 2(kd - m) ~ k 2( (n - k)d - m) and certainly only when kd - m rv n ( (clog d)21d). In fact, kd - m = 0 (n(log2 d)1d) is sufficient for our purposes. Fixing such a value of m, using n! = (nletnO(l) and M(r) = 8( (rler/ 2 ) , and noting that

(~) = exp (0 (n(log2 d)1d)) , d- c + O( d- llog2 d)),

(1 + kln)d - 1 = dC (1

-

(~)m =

(fr (fr exp(O(n(log

(f)-(kd-m)

d

d

=

3d)ld)),

((kd - m)le) kd-m = n kd- m exp (O( n(log3 d)ld)), ((n - k)d -

m) ((n-k)d-m)/2

= (nd - 2kd + (kd _ m)) ((n-k)d-m)/2

<":::

2k = (nd) ((n-k)d-m)/2 1 __

+k_m

n

2n

(

= (nd)((n-k)d-m)/2 exp ( - dk + o( n(log2 d)ld)) we find everything in the upper bound for A cancels except for

Since dC = ekd/n and the power of d is absorbed in the error term, this equals n 0(1) ( 1 - d- C + 0 (d- l log3 d) which, if c < 1, tends to 0 for large d as required. On the other hand, for c > 1 and large d, nc(log d)1d is an upper bound on the minimum dominating set size in all d-regular graphs [4, Theorem 2.2].

r,

An explicit example can be given as well: if d = (p - 1)12 with p being a prime, consider the bipartite graph H with two classes of vertices Al = A2 = Zp in which aibj forms an edge iff (ai - bj) is a quadratic nonresidue. A simple consequence of Weil's Theorem (see, e.g., [2], Section 4) implies that for every set S of at most, say, ! log2 p elements of Zp, there are more than JP members z of Zp so that (z - s) is a quadratic residue for all s E S. This implies that any dominating set of H must contain either more than! log2P vertices of A 2 or at least JP vertices of AI, and is thus of

14

N. Alon and N. Wormald

size bigger than ~ log2 p. (This can in fact be improved to (1- 0(1)) log2 p, but as we are not interested in optimizing the absolute constants here and in the rest of the paper, we omit the proof of this stronger statement). For degrees d that are not of the form (p - 1)/2 for a prime p one can take any spanning d-regular subgraph of the graph above with the smallest p for which (p - 1)/2 ~ d. Such a subgraph exists by Hall's theorem, and any dominating set in it is also dominating in the original (p -1)/2-regular graph , hence it is of size at least ~ log2 p. By the known results about the distribution of primes this prime p is (2+0(1)) d, and we thus get ad-regular graph on at most n = (4 + o(1)) d vertices in which every dominating set is of size greater than ~ log2 p > ~ log2 d ~ n(n(log d)/ d) . This completes the proof. _

3.

THE PROOF OF THE MAIN RESULT

In this section we prove Theorem 1.1. The idea of the proof is based on that of Theorem 1.2. Given a graph G = (V, E) with minimum degree d, we wish to define a dominating set C C V whose members will form the centers of the star factor , and then to assign many leaves to each of them. The trouble is that here we cannot pick the set of centers randomly, as our graph may contain a large set R of vertices of degree d whose total number of neighbors is much smaller than IRI, and then the number of centers in R is limited. This may happen if some or all of the neighbors of the vertices in R have degrees which are much higher than d. Thus, for example, if our graph is a complete bipartite graph with classes of vertices Rand U, with IRI = n - d and lUI = d « n - d, it is better not to choose any centers in R. In fact, it seems reasonable in the general case to force all vertices of degree much higher than d to be centers, and indeed this is the way the proof starts. However, if we then have a vertex all (or almost all) of whose neighbors have already been declared to be centers, then this vertex cannot be a center itself, and will have to be a leaf. Similarly, if almost all neighbors of a vertex are already declared to be leaves, then this vertex will have to become a center. The proof thus proceeds by declaring, iteratively, some vertices to be centers and other vertices to be leaves. At the end, if there are any vertices left, we choose a small subset of them randomly to be additional centers .

15

High Degree Graphs Contain Large-Star Factors

The Local Lemma has to be applied to maintain the desired properties that will enable us to apply Hall's theorem at the end to a bipartite graph, defined in a way similar to that in the proof of Theorem 1.2. An additional complication arises from the fact that we have to assign time labels to vertices and use them in the definition of the bipartite graph. We proceed with the detailed proof.

Proof of Theorem 1.1. Let G = (V, E) be a graph with minimum degree d. We first modify G by omitting any edge whose two endpoints are of degree strictly greater than d, as long as there is such an edge. We thus may and will assume, without loss of generality, that every edge has at least one endpoint of degree exactly d. Put h = lod4 / 3/(log d)I/3, and let H (for High) denote the set of all vertices of degree at least h. Since each of their neighbors is of degree precisely d, we can apply Hall's theorem and assign a set of h/d neighbors to each of them, so that no vertex is assigned twice. Let S' denote the set of all the !H!· l~dl/3 /(log d)I/3 assigned vertices, and let S (for Special) be a random subset of S' obtained by choosing each member of S' to be in S randomly and independently with probability 1/2. Claim 3.1. In the random choice of S , with positive probability the following conditions hold: (i) For each v E H , Nc(v) n > lsd1/3/(logd)I/3.

I

(ii) For each v

E

SI

V \ H, 1 NG(v) \

SI

~ d/3.

Proof. For each vertex v E H let A v be the event that condition (i) is violated for v. Similarly, for each vertex v E V \ H let B; be the event that condition (ii) is violated for v. By the standard known estimates for binomial distributions, the probability of each event A v is exp ( - n( d1/ 3/(log d)I/3)) and that of each event B; is e-O(d). In addition, each event A v is independent of all other events except for the events B u for vertices u that have a neighbor among the lod1/ 3 / (log d) 1/3 vertices of S' assigned to v (note that there are less than d4 / 3 such vertices u). The same reasoning shows that each event B; is mutually independent of all other events Au, B w with the exception of at most hd < d3 events. The desired result thus follows from the Local Lemma (with a lot of room to spare). This completes the proof of the claim. Fix an S satisfying the assertion of the claim, and define G' = G - S = (V' , E'). Note that by the above claim, part (ii),

(1)

for each v E V(G') = V \

s,

I NG'(v) 1

~ d/3.

16

N. Alan and N. Wormald

Note also that by part (i) of the claim, each vertex v E H has a set of at least 215 d1/ 3 / (log d)113 vertices from S assigned to it , and can thus serve as a center of a star of at least that size. We now construct two sets of vertices 0, LeV \ S. The set 0 will consist of vertices that are declared to be centers, and will serve as centers of stars in our final star factor . The set L will consist of vertices that are declared to be leaves in the final factor. Note that the vertices in S will not form part of these sets; they will also be leaves in the final star factor, and their associated centers will be the vertices in H to which they have been assigned, but since we have already specified their centers we do not need to consider them any more. Initially, define C = Hand L = 0. We will also need a time label t( v) which will be defined in the following for each vertex in V \ (H US) ; in the beginning set t = O. Put V' = V(G') = V\S and define D = d2/ 3(logd)1 / 3 . Now repeatedly apply the following two rules to define additional centers and leaves, and assign them time labels.

I

• (a) If there is a vertex v E V' \ (0 U L) such that NG/(v) \ 01 :s; D, add v to L, increase t by 1, and define t(v) = t.

I

• (b) If there is a vertex v E V' \ (0 U L) such that NG/(v) \ LI :s; d/6, add v to C, increase t by 1, and define t(v) = t. The process continues by repeatedly applying rules (a) and (b) in any order until there are no vertices left in V' \ (0 U L) that satisfy the conditions in rule (a) or in rule (b). Let to denote the value of the time parameter t at this point . Actually, since Land C will be disjoint, by (1) no vertex will satisfy the conditions in both rules simultaneously. Let F (for Free) denote the set of all vertices in V' \ (C U L) remaining once the process terminates. Define p = 20(log d)/ d, and let T be a random subset of F obtained by picking each vertex v E F, randomly and independently, to be in T with probability p. Assign the vertices of F \ T the time labels to + 1, to + 2, . . . , to + IF \ TI in any order. Finally, assign the vertices of T the time labels to + IF \ TI + 1, to + IF \ TI + 2, ... , to + IFI. In our final star factor, the vertices HUOUT will serve as centers, while the remaining vertices, that is, those in S U L U (F \ T), will serve as leaves. In order to show that it is possible to define large stars with these centers and leaves, we need the following.

High Degree Graphs Contain Large-Star Factors

17

Claim 3.2. With positive probability, every vertex of F has at least one neighbor in CUT, and no vertex v E V' - H has more than 2ph = 4d1/3(log d)2/3 neighbors in T. Proof. For each vertex v E F that does not have any neighbor in C, let A v be the event that it has no neighbor in T. Note that as v E F, the definition of rule (b) implies that Nc'{v) \ LI > d/6 , and as it has no neighbor in C, it has more than d/6 neighbors in F. Therefore, the probability that none of these neighbors is in T is at most (1 - p)d/6 < d- 3. For each vertex v E V' \ H, let B; be the event that v has more than 2ph neighbors in T. Since the degree of v in G' is at most h, its number of neighbors in F is certainly at most h, and hence the standard estimates for binomial distributions imply that the probability of each event B; is at most e-O(ph) which is much smaller than, say, d- 3 .

I

Note that each event A v is mutually independent of all other events Au or B w apart from those corresponding to vertices U or w that have a common neighbor with v in F , and the number of such vertices u, w is smaller than hd < d7 / 3. Similarly, each of the events B; is independent of all others but at most hd < d7 / 3. The claim thus follows from the Local Lemma. Returning to the proof of the theorem, fix a choice of F satisfying the assumptions in the last claim. Let B be the bipartite graph with classes of vertices (C \ H) U T and L U (F \ T), in which each v E (C \ H) U T is adjacent to any of its neighbors u that lies in L U (F \ T) and satisfies t{u) < t(v). Note that, crucially, prospective centers are connected in B only to prospective leaves with smaller time labels. Our objective is to show, using Hall's theorem, that we can assign to each vertex v in (C \ H) U T some O{d1/ 3 /(log d)1/3) neighbors of v (in B, and hence also in G') from LU{F\T) , such that each vertex in LU{F\T) is assigned at most once. To do so, we first establish several simple properties of the bipartite graph B that follow from its construction. Claim 3.3. The following properties hold.

INB(U) n (O \ H) I : :; D = d2/3{10g d)1/3. (ii) For each vertex u E L, INB{u) n TI :::; 4d1/3{logd)2/3. (iii) For each vertex u E F \ T, INB{U) I = INB{U) n TI :::; 4d1/3{10gd)2/3. (i) For each vertex u E L,

(iv) For each vertex v E 0 \ H , dB{v) 2: d/6.

(v) For each vertex vET, INB{V)! 2: D - 4d1/3{logd)2/3 > D/2 = !d2/3{logd) 1/3 .

18

N. Alon and N. Wormald

Proof. (i) By the definition of rule (a), each u E L can have at most D neighbors with time labels exceeding t(u) , and therefore can have at most that many neighbors in B .

(ii) This follows immediately from the condition in Claim 3.2 that F was chosen to satisfy. (iii) By the definition of the graph B, the vertices in F \ T are joined in B only to vertices of T , as these are the only vertices with bigger time labels. Therefore, 1 NB(U) I = I NB(u) n TI for each u E F \ T, and the claimed upper estimate for this cardinality follows from Claim 3.2. (iv) By the definition of rule (b), each vertex v E C \ H satisfied ING'(v) \ at the point of being added to C. Since by (1), 1NG,(v)1 ~ d/3, it follows that at that time , v had at least d/3 - d/6 = d/6 neighbors in L. As all these leaves have smaller time labels than v, it is joined in B to all of them.

£1 :::; d/6

I

(v) If vET, then v E F, and thus, by the definition of rule (a), NG'(v) \ > D. Vertices in T are given the largest time labels , so in the graph B, v is joined to all members of N G/ ( v) \ 0 except for those that lie in T. However, by the condition in Claim 3.2, at most 4d1/ 3(logd)2/3 of these vertices are members of T, implying the desired estimate. This completes the proof of th e claim.

CI

Corollary 3.1. For each subset X d1/ 3 / (log d) 1/3 .

c

(0 \ H) U T, INB(X)I

> IXI·

l6

Proof. If at least half the elements of X belong to C \ H, then, by Claim 3.3(iv), the total number of edges of B incident with them is at least ~IXI . id. By the first observation in the proof of part (iii) of the claim , these edges are not incident in B with any member of F \ T. By part (i) of the claim, at most D = d2/ 3(log d)1 /3 of these edges are incident with any one vert ex in L. Thus, in this case, IN(X)I ~ ~IX I' id.1/D = IXI ·l d1/ 3/ (log d)1/3, providing the required estimate. 6

Otherwise, at least half of the vertices of X lie in T . By Claim 3.3, part (v), the total number of edges of B incident with them is greater than ~ IXI ·~D. By parts (ii) and (iii) of the claim, each neighbor of these vertices in LU(F\T) is incident with at most 4d1/ 3(logd)2/3 of these edges, implying

19

High Degree Graphs Contain Large-Star Factors

that in this case

N (X ) I >1!lD

I

-

2

1 2 4d1/3(log d)2/3

d 1/ 3

16(logd)1/3

IX1.

This completes the proof of the corollary. By the last Corollary and Hall's theorem, one can assign a set of d1/ 3 / (log d)1/3 members of L U (F \ T) to any element of (C \ H) U T, so that no member of L U (F \ T) is assigned more than once. This makes all elements of (C \ H) U T centers of large vertex disjoint stars. Adding to these stars the stars whose centers are the elements of H and whose leaves are those of 5, we may apply Claim 3.2 to conclude that if there are any unassigned vertices left in L we can connect each of them to one of the existing centers. Similarly, (1) and the definition of rule (a) do the same job for unassigned vertices in F \ T. Thus, we get a star factor in which each star has at least 215 d1/ 3 / (log d) 1/3 leaves. This completes the proof. •

l6

4.

CONCLUDING REMARKS AND OPEN PROBLEMS

We have shown that for every positive integer 9 there is an integer d so that any graph with minimum degree at least d contains a star factor in which every component has at least 9 edges. Let d(g) denote the minimum number d for which this holds. Our main result shows that d(g) O(g3log g), while the construction described in the proof of Theorem 1.2 implies that d(g) 2: n(g log g). It seems plausible to conjecture that d(g) = 8(g log g), but this remains open. It will be interesting to determine d(g) precisely (or estimate it more accurately) for small values of g, like 9 = 2 or 3. Our proof, even if we try to optimize the constants in it, will yield only some crude upper bounds that are certainly far from being tight. It is worth noting, however, that even for showing that d(2) is finite, we do not know any proof simpler than the one given here for the general case. On the other hand, a random 3-regular graph contains a Hamilton cycle with probability tending to 1. Hence, if it has number of vertices divisible by 3, it contains a spanning factor of stars of two edges each. Similarly, a random 4-regular graph having number of vertices divisible by 4 contains a spanning factor

:s

of stars of three edges each with probability tending to 1 [5]. Immediately

20

N. Alon and N. Wormald

from contiguity results discussed in [12], the same statements are true if we change 4-regular to d-regular for any d ~ 4. Our proof, together with the algorithmic version of the local lemma proved by Beck in [6] (see also [1]) , and any efficient algorithm for bipartite matching, show that the proof here can be converted to a deterministic, polynomial time algorithm that finds, in any given input graph with minimum degree at least d, a star factor in which every star is of size at least n( (d/logd)1 /3). We omit the details. There are several known results that show that any connected graph with large minimum degree contains a spanning tree with many leaves, see [11], [8], [7]. In particular, it is known (and not difficult) that any connected graph with minimum degree d and n vertices contains a spanning tree with at least n - 0 ( n(log d)/ d) leaves. A related question to the one considered here is whether it is true that any connected graph with large minimum degree contains a spanning tree in which all non-leaf vertices have large degrees. Specifically, is there an absolute positive constant c so that any connected graph with minimum degree at least d contains a spanning tree in which the degree of any non-leaf is at least cd/ log d? Another intriguing question is the following possible extension of the main result here. Is it true that the edges of any graph G with minimum degree d can be partitioned into pairwise disjoint sets, so that each set forms a spanning star forest of G in which every component is of size at least h(d), where h(d) tends to infinity with d? A related result is proved in [3], but the proof of the last statement, if true, seems to require additional ideas.

REFERENCES

[1]

N. Alon, A parallel algorithmic version of the local lemma, Random Structures and Algorithms, 2 (1991), 367-378.

[2J

N. Alon, Tools from higher algebra, in: Handbook of Combinatorics (R. L. Graham, M. Grotschel and L. Lovasz, eds) , North Holland (1995), Chapter 32, pp. 17491783.

[3]

N. Alon, C. McDiarmid and B. Reed, Star arboricity, Combinatorica , 12 (1992), 375-380.

[4J

N. Alon and J. H. Spencer , The Probabilisti c Method, Third Edition, Wiley (New York, 2008).

[5J H. Assiyatun and N. Wormald, 3-star factors in random regular graphs , Europ. J. Combinato rics, 27 (2006), 1249-1262.

21

High Degree Graphs Contain Large-Star Factors

[6]

J. Beck, An algorithmic approach to the Lovasz Local Lemma, Random Stru ctures and Algorithms, 2 (1991),343-365 .

[7J Y. Caro , D. B. West and R. Yuster, Connected domination and spanning trees with many leaves, SIAM J. Discrete Math., 13 (2000), 202-211. [8J J . R. Griggs and M. Wu, Spanning tre es in graphs of minimum degree four or five, Discrete Mathematics, 104 (1992), 167-183 . [9]

F. Havet, M. Klazar, J. Kratochvil , D. Kratsch and M. Liedloff, Exact algorithms for L(2, I)-labelling, Algorithmica, to appear.

[10]

J . Kratochvil, An open problem presented in "Building Bridges" , Budapest, 2008.

[11] D. J. Kleitman and D. B. West, Spanning trees with many leaves, SIAM Journal on Discrete Mathematics, 4 (1991), 99-106 . [12]

N. C. Wormald, Models of random regular graphs, in: Surveys in Combinatorics, 1999, London Mathematical Society Lecture Note Series 267 (J. D. Lamb and D. A. Preece, eds), Cambridge University Press (Cambridge, 1999), pp . 239-298 .

Noga Alon

Nicholas Wormald

Tel Aviv University Tel Aviv 69978 Israel and lAS, Princeton, NJ 08540, USA

Department of Combinatorics and Optimization University of Waterloo Waterloo ON Canada

e-mail:

nogaa~tau.ac.il

e-mail:

nwormald~waterloo.ca

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp . 23-42.

ITERATED TRIANGLE PARTITIONS

STEVE BUTLER* and RON GRAHAM

For a given triangle there are many points associated with the triangle that lie in its interior; examples include the incenter (which can be found by the intersection of the angle bisectors) and the centroid (which can be found by the intersection of the medians) . Using this point one can naturally subdivide the triangle into eit her three or six "daughter" triangles. We can then repeat the same process on each of the daughters and so on and so on. A natural question is after some large number of steps what does a typical nth generation daughter look like (up to similarity)? In this pap er we look at this problem for both the incenter and the centroid and show that they have very distinct behavior as n gets large . We will also consider the Gergonne point and the Lemoine point.

1.

INTRODUCTION

In this paper we will be considering triangles and are concerned only with the shape, i.e., up to similarity. So we can denote a triangle T by the triple T(A, B, C) where A, Band C are the interior angles. Given a point P in the interior of T we can use the point to subdivide T into smaller "daughter" triangles. This is done by taking each vertex and drawing the line passing through P and connecting it to the opposite side (sometimes referred to as Cevians), subdividing the triangle into six daughters (Figure la); or by taking each vertex and drawing the line to P subdividing into three daughters (Figure 1b). We are interested in the case when P is a well-defined point lying in the interior of T for any non-degenerate T . Examples of such points include the "This work was partially done with support of an NSF Mathematical Sciences Postdoctoral Fellowship.

24

S. Butler and R. Graham

(a) Six daughters

(b) Three daughters

Fig. 1. Given an interior point P how to subdivide the triangle

incenter (which is the intersection of the angle bisectors of T), the centroid (which is the intersection of the medians of T), along with many others. A rather complete listing of well known distinguished points associated with triangles is maintained at the Encyclopedia of Triangle Centers [9] (however not all of these points will always lie in the interior of T, the excenter being one such example) .

Once we have settled on a way of choosing P, we can iterate the process of subdivision, say for n times, producing 6n (or 3n ) nth generation daughters from our original triangle T. We are interested in what can be said about an nth generation daughter of T as n goes to infinity. For example, what is the distribution of the shape of the daughters? What is the distribution of the smallest angle (or second smallest angle) of T? It will turn out that the answers to these questions depend in a crucial (and currently not well understood) way on exactly how P is chosen. In the following sections, we will specify various choices for P and address these questions and show that well known points P can produce dramatically different results as n goes to infinity. To help us answer these questions we first will need a convenient way of describing the triangles. Each triangle T(A, B, C) corresponds to a point (A, B, C) E ]E3 which will lie in the intersection of the plane x + y + Z = 1f with the positive orthant (see Figure 2). We will denote this intersection, which is actually an equilateral triangle, by P (this representation has been used before in the analysis of pedal triangles; see [1, 10, 11]).

2. SUBDIVIDING BY BISECTORS One of the easiest points P to work with is the incenter which is formed by using the angle bisectors. What is nice about this point is that the angles in

25

Iterated Triangle Partitions

(0. IT . 0)

Fig. 2. Representing triangles as points in lE3

the daughter triangles are linear combinations of the angles in the original triangle. So the angles of the daughter triangle can be formed via matrix multiplication. This transforms the problem into a Markov process using matrices which has been extensively studied (see the survey [6] for more information). We will consider both possibilities of subdivision shown in Figure 1.

2.1. Subdividing into three daughters If we associate the triangle T( A, B, C) with the vector t = (A, B, C) * in ]E3, then the three daughters of T can be found by Mit where i E {1, 2, 3} and

1/ 2

M1

0 0)

= 1/2 1 1/2 , (

o

0 1/2

M2

1 1/2 1/2)

= 0 1/2 (o

0

0

1/2

,

M3

=

1/ 2 (

0

0

1/2

0)

0 .

1/2 1/2 1

The nth generation daughters of T can then be found by Mil M i2 .. . Mint where each ij E {1,2,3}. Starting with the triangle T(1r/9, 21r/9 , 21r/3) we have plotted the nth generation daughters for n = 2,4,6 ,8 in Figure 3. The figures are very suggestive about what is happening. To see why we are getting the Sierpinski triangle it helps to see how the image of M, maps P to P . This is shown in Figure 4. So we have that there will be exactly one daughter in each of the subtriangles indicated in Figure 5a. Similarly, if we look at the location of the second generation daughters there will be one daughter in each of the nine triangles indicated in Figure 5b, and if we look at the location of the third generation daughters there will be one daughter in each of the twenty seven triangles indicated in Figure 5c. Since this process mirrors that

s.

26

(a) n

(b) n

= 2

=4

Butler and R. Graham

(c) n = 6

(d) n

=8

Fig . 3. The nth generation daughters for angle bisectors with three daughters at each

generation

Fig .

4. The image of P under the three maps

M,

used to form the Sierpinski triangle, as n goes to infinity the distribution of daughters are the points in the Sierpinski triangle.

(a) First generation

(b) Second generation

(c) Third generation

Fig. 5. Location of the descendants in the first three generations

2.2. Dividing into six triangles As before if we associate a triangle T(A, B, C) with the vector t = (A, B, C)* in ]E3, then the six daughters of T can be found by Mit where i E {I, 2, 3, 4, 5, 6} and

M1 =

1/ 2

0

0)

1/2 1/2 0 , 1/2 1

( o

1/2 M2 =

0

( 1/2

o 1/2) 1/2 o , 1/2

1/2

27

Iterat ed Triangle Partitions

1 1/2

M4 =

0 1/2 ( o 0

0)

1/2 , 1/2

M5 =

1/ 2 0 1/2) 1/2 1 0 , ( o 0 1/2

M6 =

1/ 2 0 0 1 ( 1/2 0

And as before we can form the nth generation daughters by looking at all 6n possible products of the form M il M i2 ... M in t with each ij E {l , 2, 3, 4, 5, 6}. Starting with the equilateral triangle T 1 = T(7r /3, 7r /3, 7r /3) we have plotted the nth generation daughters for n = 1,3 ,5 in Figure 6.

... ... .

(a) n

=1

.. .......

(b) n

=3

(c) n

=5

Fig. 6. The nth generation daughters for angle bisectors with six daughters at each generation

Looking at Figure 6c we see that the daughters seem to fill in Prather uniformly. A patient count though will reveal that there are far fewer than 65 triangles in Figure 6c. It is not hard to see that an nth generation daughter of the equilateral triangle must have the form T( Ul7r /2 n , U27r /2 n , U37r/2 n ) for positive integers Ul , U2 , U3 with Ul + U2 + U3 = 2n . In particular there are at most (2;) possible different daughter triangles among the 6n which will be generated. So on average each triangle is being hit approximately 2(3/2t times. In particular, any triangle which is not being hit must be missed for a good reason. It is not too hard to see that one condition that is needed is for gcd(Ul,U2,U3) = 1, which rules out a positive fraction of the daughters but does not still explain all of the missing daughters. If we start with a triangle which does not have rational multiples of 7r then we do not have as many triangles stacking up on top of each other. As an example , the fifth generation daughters of T2 = T( 7r /3, V27r /3,

( 2 - V2) 7r /3) are shown in Figure 7 (this should be compared to Figure 6c). One problem that we run into is that the triangles can be fairly densely packed in these pictures. So instead of plotting the individual triangles it is better to look at a histogram. We will divide P into a large number of small

28

S. Butler and R. Graham

Fig . 7. The fifth generation daughters for angle bisectors starting with an "irrational"

triangle

regions and then shade each region according to the number of triangles that fall into that region, the darker a region is the more triangles fall into that region. We have plotted histograms for Tl and T2 in Figure 8 (these should be compared with Figures 6c and 7). The situation is obviously much richer than the previous case when we looked at three daughters. The reason for this is previously the image of each one of the M, was disjoint, but this is no longer the case. If we plot the images of the six maps we get Figure 9, in particular there is a lot of overlap between pairs of maps. However, we still have one nice feature of these maps . Namely, they are contracting. To see this we note that we can put Pinto ]E2 by putting T(A,B,C) at ((A + 2B)jV3,A) . We then can put the first generation daughter T(Aj2,(A+B)j2,(B+2C)j2) at ((3A+2B)j(2V3),A/2). If we now compare the distance between triangles T(A, B, C) and T(A' , B' , C') and their daughters, a calculation shows

~ ((A + 2B

_ A'

_ ((3A

+ 2B

4

V3

2V3

+ 2B')2 + (A _

V3

_ 3A'

A')2)

+ 2B')2 + (A

2V3

2

_ A')2) =

2

~(B _ B,)2 > O. 3

-

It follows that for the euclidean distance d we have

d(T( A/2, (A

+ B)/2, (B + 2C)/2) , T( A' /2, (A' + B')/2, (B' + 2C')/2)) :::;

~ d(T(A, B, C), T(A', B' , C')) .

29

Iterated Triangle Partitions

(a) Triangle T1 ; n

=5

(b) Triangle T2 ; n

=5

(c) Triangle T1 ; n

= 12

(d) Triangle T2 ; n

= 12

Fig. 8. Histograms for the distribution of triangles in P under bisector division

Fig. 9. The image of P under th e six maps M,

By symmetry, the same statement holds for all the daughter triangle maps. Since these maps are contracting with Lipschitz constant ..;3/2 then it follows (see [6]) that there is a fixed stationary distribution on P that the process converges to . Further it converges exponentially. Hence Figures 8c and 8d are nearly identical, and these are approximations for the histogram of the limiting distribution.

30

S. Butler and R. Graham

2.2.1. The smallest angle. We now consider the problem of analyzing the distributions of the smallest angles. Just as we have a quick rate of convergence to the stationary distribution in P: we experimentally see a quick convergence to the same distribution of minimal angles. In Figure 10 we split the interval between a and 1r/3 into 1000 intervals and took 50, 000, 000 random walks of length 50 in this Markov process and recorded the minimum angle.

Fig . 10. The distribution of th e smallest angle using bisectors

A few things certainly stand out . Perhaps the simplest thing to observe is that the minimum angles are not concentr ated near 0, so that the nth generation daughters are not becoming "flat" . There also is a certain amount of self-similarity (but given the distributions we saw in Figure 8cd this should not be surprising). We also have marked a few of the values between a and tt /3 in Figure 10. In particular we see that a few of the "valleys" are located at 1r/16,1r/8, 31r/16, 1r/4 and a few of the peaks are located at tt /10, 31r/20, 71r/ 40, 1r/5. The tallest peak is located at 1r/5. If we look closely at the histograms in Figure 8cd we see that the darkest region is located at T(1r/5,1r/5, 21r/5). This particular triangle has the following unique property; T( 1r/5, 21r/5, 21r/5) is the only triangle where two of its daughters are similar to itself (this is shown as the shaded triangles in Figure 11). In terms of the M, this means that some permutation of the vector t = (1r/5, 21r/5 , 21r/5)* is an eigenvector associated with eigenvalue 1 for two of the Mi. The only other nondegenerate triangle that also acts as an eigenvector is T(21r /9, tt /3, 41r/9) which has one of its daughters similar to itself. Figure 10 was computed using random walks and so the distribution is at best approximate. It is not too difficult to find upper and lower bounds for the distribution (though in practice the amount of computation needed

31

Iterated Triangle Partitions

Fig. 11. The decomposition of T(1r/5, 21r/ 5, 21r/ 5)

to compute these bounds to an accuracy to be able to produce Figure 10 is prohibitive) . The basic idea is to note for a fixed k that Mil ... M ik maps P into a smaller triangular region of P , let us denote this by Q. SO in particular we have that Mil ' " MikMik+l ... Min t must lie in Q as well for any choice of initial t and any choice of the ik+l,"" in . In essence if we know what the last k steps of this Markov process are then we have a lot of information about where in P we end up (the higher the value of k the more precisely we can approximate the point). With this in mind the procedure is to fix a value of k and then look at the image of P under each of the 6k possible combinations of Mil " ·Mik . Now divide the interval between 0 and 1f/3 in some fashion. For each interval a lower bound for the distribution is found by counting the number of images Q which must have smallest angle in that interval divided by 6k ; an upper bound is found by counting the number of images of Q which can have some smallest angle in the interval divided by 6k . An example of the resulting bounds on the distribution is shown for k = 15 and the interval split into widths of tt /256 in Figure 12.

.

:I

Fig. 12. Upper (black) and lower (red) bounds for the true distribution of smallest angle

The precise location of the peaks and valleys is in general not well understood. For comparison we also include the distribution of the middle

32

S. Butler and R. Graham

and largest angles Figure 10.

III

Figures 13 and 14 computed in the same way as 1T

o

"2

Fig. 13. The distribution of the middle angle using bisectors

Fig. 14. The distribution of the largest angle using bisectors

3.

SUBDIVIDING BY MEDIANS

We now consider the subdivision using the centroid, or median point of the triangle. We will restrict our attention to the case of six daughters. This case has been well studied because of its close relationship to barycentric subdivision. We will see that this situation is quite different from the case of the incenter. One major difference is that the median map is not contracting on P as we saw previously. In Figure 15 we indicate how one of the median daughters maps P to itself (the other five are similar and differ by rotation and reflections). To get some intuition about what is happening in this case let us consider the histogram for T( 7f /3, 7f /3, 7f /3) for n = 4,6,8,10 shown in Figure 16.

33

Iterated Triangle Partitions

Fig . 15. A subdivision of P and its image under th e median map

(a) n

=4

(b) n

=6

(c) n

=8

(d) n

= 10

Fig . 16. The nth generation daughters for medians with six daughters at each generation

Looking at these pictures what seems to be happening is that the triangles in P tend to go towards the vertices as n increases. Or in other words, the triangles are becoming "flat" , or nearly colinear, as n gets large. This behavior was noted by David Blackwell and was passed on to us by Persi Diaconis [5] , independently Barany et al. [2, 15], inspired by a question of Stakhovskii about the distribution of shapes of triangles, showed that most triangles are flat.

3.1. Analytic approach

Blackwell started by putting a triangle in a standard position by putting the longest side along the x-axis with vertices at (0,0) and (1,0), the third vertex is then put with y > 0 and x ~ 1/2. Given a triangle T in standard position we now define the pseudo-fatness of T, denoted PF(T), by PF(T) = JY(3 - x) . It should be noted that for a triangle in standard position that y /2 is the area of the triangle, so the term JY relates to the area. The term (3 - x) acts as an error correction term. If T 1 , . .. , T6 are the six daughter triangles of T then Blackwell noted that if the average of the pseudo-fatness of the daughters was small compared to

34

S. Butler and R. Graham

the pseudo-fatness of T for each triangle T, i.e.,

~ LPF(Td ::; cPF(T) t:

for all T and some c < 1, then most ofthe triangles will have pseudo-fatness going to zero. In particular, most of the triangles would be flat. Computation using Maple confirms this assertion for the value c = 0.99 as we show in Figure 17. From this, for example, it would follow that the number of nth generation daughters with smallest angle greater than 0.99n radians is at most 5.9n . However, at present no one has confirmed analytically that the above inequality is valid with this value of c. Also it would be interesting to find the best pseudo-fatness function.

0.0

0.1

0.4

0 .6

0.3

1.0

0 .9

0.1

0.7

0 .6

OS

Y

Fig . 17. Average of the pseudo-fatness of daughter triangles, seen from the side

There are weaknesses in this approach . Namely, while we can show that the smallest angle is small, this does not automatically imply that the largest angle is large (~ 1r) . Or put another way, this approach shows that the triangles drift to the edges of P but not necessarily to the vertices of P. Robert Hough [8J using a different technique was able to show that the largest angle does in fact approach 1r and moreover gave asymptotic bounds for the proportion of triangles with angles near 1r, this shows that the triangles do accumulate at the vertices of P.

35

Iterated Triangle Partitions

3.2. Hyperbolic approach The method of Barany et al. was to move the problem from P to the hyperbolic half plane. The following approach is similar to theirs and was discovered independently by Curt McMullen [14]. The first step is to associate with each triangle T (up to) six points z in the hyperbolic upper half plane lHI as shown in Figure 18. Namely, some edge of T is located with vertices at z = 0 and z = 1 and the third vertex is located at the complex coordinate z with positive imaginary part. McMullen then observes that reflecting z across the three circles R(z) = ~' [z] = 1, and [z - 11 = 1 induces a natural action of 8 3 on lHI in which all six orientations of T occur.

R(z) = &

[z] = 1

Fig . 18. Triangles as points in the hyperbolic plane

Now, the centroid point of a triangle in this position is represented by the point (z + 1)/3, and so one of the corresponding daughters becomes 2(z + 1)/3 when normalized. The group of automorphisms of !HI generated by the map B(z) = 2(z + 1)/3 and 83 is dense in Aut(lHI) (see [2]). From this, using results of Furstenberg [7] , it follows that almost all random walks formed from products of B(z) and elements of 8 3 tend to infinity (in the hyperbolic plane) as the length of the product increases. This then implies that almost all of the nth generation daughters have smallest angle tending to 0 as n increases. Actually more can be said in that these points are dense in the hyperbolic space and so every triangle is arbitrarily "close" so some descendant of the starting triangle.

36 4.

S. Butler and R. Graham SUBDIVIDING BY THE GERGONNE POINT

We again restrict ourselves to the case of six daughters. The Gergonne point of a triangle is the point of concurrence of the three line segments joining each vertex of the triangle to the point of tangency of the inscribed circle to the side opposite the vertex. We show an example of this in Figure 19. (As a side note the example shown is the only triangle (up to similarity) with the equilateral triangle as one of the daughters; its side lengths are proportional to 19, 40 and 49, see [4] .)

Fig. 19. A triangle and its Gergonne point G

In Figure 20 we look at how one of the Gergonne daughters maps P to itself. It is interesting to note that unlike the medians or angle bisectors this map is not 1-to-l. That is, there is a small region which gets mapped 2-to-l.

Fig . 20. A subdivision of P and its image under the Gergonne map

Looking at Figure 20 it also looks like the map might be contracting

(this would then imply a limiting distribution). However this is not the case. For instance if we consider the following two triangles in the corner T 1 (170°,5°,5°) and T 2(160°, 10°,10°) then two of their corresponding

37

Iterated Triangle Partitions

daughters are respectively DT1 (170.0377.. .° ,9 .9245...° ,0.0377 ... 0) and DT2(160.2935 . .. 0 ,19.4129 ...°,0.2935 ... 0). SO IIDT1 - DT211/IITl - T211 = 1.1106... showing that there is a slight expansion in the corner. This does not rule out a limiting distribution (and the authors do believe that a limiting distribution exists) it only shows that more sophisticated techniques will be needed. In Figure 21a we have drawn the histogram for the tenth generation daughters (higher generation daughters seem to have a similar structure) . There appears to be a fair amount of white, this is caused not by there being very few triangles in the region, but no triangles in the region. In Figure 21b we have darkened the histogram by making any region with a daughter black and any region without a daughter white. As the figure indicates there seems to be large regions with missing daughters. It would be interesting to know what triangles are possible for the nth generation (this is not even known precisely for the first generation) .

(a) Histogram

(b) Darkened

Fig . 21. The tenth generation daughters for the Gergonne point with six daughters at

each generation

In Figure 22 we indicate the distributions for the middle and largest angles using the Gergonne point divison. Looking at the distributions it seems that the middle angle is concentrated around 1r /3 while the largest angle has some approximate symmetry around 1r /2 . For the largest angle what appears to be happening is that for "most" triangles when using the Gergonne point the largest angles of the daughters are located at the vertex along the edge of the parent triangle. So by pairing up adjacent daughters along the edge their two largest angles should be symmetric around rr/2.

38

S. Butler and R. Graham

" 3"

o

l . . - - -_

_

7T

"2

7T

"2

J ~----...J

(a) Middle angle

(b) Largest angle

Fig . 22. The distribution of angles using the Gergonne point

5.

SUBDIVIDING BY THE LEMOINE POINT

The Lemoine point of a triangle is the intersection of the symmedians which are the medians reflected across the angle bisectors, so it is also known as the symmedian point. This point seems to be similar to the medians in that the smallest angles go to 0 but experimentally the triangles seem to be going to the sides of P rather than the corners of P. Further, they seem to be drifting slowly. In Figure 23a we have plotted the histogram for all the 10th generation daughters while Figure 23b was formed by random walks of length 30 using the Lemoine point to subdivide.

(a)n=lO

(b) n

= 30

Fig. 23. The nth generation daughters for the Lemoine point with six daughters at each

generation

Iterated Triangle Partitions

39

In Figure 24 we look at how one of the Lemoine daughters maps P to itself. Clearly the map is not contracting. It is also difficult to see what is happening on the right hand side, this is because there is a region which gets mapped 3-to-1 onto. It would be interesting to know more about this mapping, for instance where is the folding occurring (experimentally it seems to be folding near the preimage point of T( 1r /3, tt /3, 1r /3)) .

Fig . 24. A subdivision of P and its image under the Lemoine map

6.

CONCLUDING REMARKS

There are still a large number of questions yet to be answered. For instance, what can be said about the limiting distribution when using bisectors? Is there a limiting distribution when using the Gergonne point, if so what is it? What is the behavior as n tends to infinity when using the Lemoine point? If the triangles are becoming flat what can be said about the rate of convergence? When looking at the medians we were interested in how the two smallest angles compared. When looking at the square of the smallest angle divided by the second smallest angle (giving a ratio between 0 and 1) we got the histogram in Figure 25 divided into 5000 slots; 13611504/17702781 went into the first slot and so we removed that extreme case. The remaining data showed spikes in unusual places, which seem to be connected to the Farey fractions. Is there any explanation for this behavior? In addition, there is of course a vast catalogue of additional points that can be considered (see [9]) . One can also consider variations such as alternating between bisectors and medians at each stage, or using as

40

S. Butler and R. Graham

a

1 -2

·9L7'1.ra o~ ·~TT' ~3 min " .:!4;;7

1

54

.1,

,

I

Fig . 25. The square of the ratio of the smallest and middle angles using 4091277 random

walks with medians of depth 300

a central point the midpoint between the incenter and centroid. A much more general question would be to ask what characterizes points where in the limit most triangles are not flat? One could also consider the problem of where we pick the point uniformly at random in each daughter. In this direction Mannion [12, 13] has shown that if we choose all three vertices at random in the triangle then almost surely the limiting process is collinear. Similar questions can be asked for higher dimensional analogues. For example Schwartz [18, 19] looked at the distribution of "shapes" when nsimplices are recursively subdivided using Barycentric subdivision. In that case he showed that the behavior is similar to the case of division of triangles using the centroid. It would be interesting to see if there were other well defined points in simplices which generate different behavior (Le. , similar to that of the incenter or Gergonne points). Beyond the int rinsic curiosity of these iterated triangle partitions there is also a vast literature on how to subdivide triangles so that the minimal angle is always bounded away from 0 (see [3, 16, 17]). This is desirable since certain methods can fail when the subdivision creates a large number of triangles with minimal angles going to 0 as n gets large. So for instance using medians for subdividing is not desirable since this creates many flat triangles. On the other hand, if we use angle bisectors we can guarantee that all but f(c) proportion of the triangles have minimal angles at least e so that we can limit the proportion of "bad" triangles.

Iterated Triangle Partitions

41

Acknowledgements. We thank Persi Diaconis and Curt McMullen for many useful discussions and insights and the referee for directing us to related literature.

REFERENCES [1]

J. C. Alexander, The symbolic dynamics of pedal triangles, Math. Mag., 66 (1993), 147-158 .

[2]

Imre Barany, Alan F. Beardon and Thomas Keith Carne, Barycentric subdivision of triangles and semigroups of Mobius maps, Mathematika 43 (1996), 165-17l.

[3]

Marshall Bern and David Eppstein, "Mesh generation and optimal triangulation" , in: Computing in Euclidean Geometry, Ding-Zhu Du and Frank Hwang, eds. (1992), 23-90.

[4]

Steve Butler, The lost daughters of Gergonne, Forum Geometricorum, 9 (2009), 19-26.

[5]

Persi Diaconis , personal communication.

[6]

Persi Diaconis and David Freedman, Iterated random functions, SIAM Review, 41 (1999), 45-76.

[7]

Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428.

[8]

Robert Hough, personal communication.

[9]

Clark Kimberling, Encyclopedia of Triangle Centers, http ://faculty .evansville.edu/ck6/encyclopedia/ETC .html

[10]

John Kingston and John Synge, The sequence of pedal triangles, Amer. Math. Monthly, 95 (1988), 609-620.

[11)

Peter Lax, The ergodic character of sequences of pedal triangles, Amer. Math. Monthly, 97 (1990), 377- 38l.

[12]

David Mannion, Convergence to collinearity of a sequence of random triangle shapes, Adv. Appl. Prob., 22 (1990), 831-844.

[13]

David Mannion, The invariant distribution of a sequence of random collinear triangle shapes, Adv. Appl. Prob., 22 (1990), 845-865 .

[14]

Curt McMullen, personal communication.

[15]

Andrei A. Ordin, A generalized barycentric subdivision of a triangle, and semigroups of Mobius transformations, Russian Math. Surveys, 55 (2000), 591-592.

[16]

Maria-Cecilia Rivar and Gabriel Iribarren, The 4-triangles longest-side partition of triangles and linear refinement algorithms, Math. of Computation, 65 (1996), 1485-1502.

[17]

Ivo Rosenberg and Frank Stenger, A lower bound on the angles of triangles constructed by bisecting th e longest side, Math. of Computation, 29 (1975),390-395.

[18J Richard Evan Schwartz , The density of shapes in three-dimensional barycentric subdivision, Discrete Comput. Geom., 30 (2003), 373-377 . [19J Richard Evan Schwartz , Affine subdivision, steerable semigroups , and sphere coverings, Pure and Applied Mathemat ics Quarterly, 3 (2007), 897-926 .

Steve Butler

Ron Graham

Department of Mathematics UCLA , Los Angeles CA 90095

Department of Computer Science and Engineering University of California San Diego, La Jolla CA 92093

e-mail:

butler~ath.ucla.edu

e-mail :

graham~ucsd.edu

BOLVAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp . 43-62 .

PAGERANK AND RANDOM WALKS ON GRAPHS

FAN CHUNG and WENBO ZHAO Dedicated to Loiuisz on the ocassion of his sixtieth birthday

We examine the relationship between PageRank and several invariants occurring in the study of random walks and electrical networks . We consider a generalized version of hitting tim e and effective resistance with an additional parameter which controls the 'speed' of diffusion. We will establish their connection with PageRank. Through these connections, a combinatorial interpretation of PageRank is given in terms of rooted spanning forests by using a generalized version of the matrix-tree theorem. Using PageRank, we will illustrate that the generalized hitting time leads to finding sparse cuts and efficient approximation algorithms for PageRank can be used for approximating hitting time and effective resistance.

1. INTRODUCTION

The notion of PageRank, first introduced by Brin and Page [2] , forms the basis for their Web search algorithms. Although the original version of PageRank was used for the Webgraph (with all the webpages as vertices and hyperlinks as edges), PageRank is well defined for any given graph and is quite effective for capturing various relations among vertices of graphs . In this paper, we will investigate several implications of PageRank for a given graph. To start with, we give the graph-theoretical definition of PageRank. Roughly speaking , PageRank is a way to organize random walk of various lengths. Instead of having to determine the number of steps a random walks is taking, PageRank uses a positive real value a , where a E [0,1) to control the "diffusion" of a combination of random walks.

44

F. Chung and W. Zhao

The original definition for PageRank was to assign a value to each vertex (Webpage), denoting the "importance" of a vertex under two assumptions: For some fixed probability a, a surfer at a Webpage jumps to a random Webpage with probability a and goes to a linked Webpage with probability I-a. The importance of a Webpage v is the expected sum of the importance of all the Webpages u that preceed v. In this paper, we will use a more general version of PageRank, called personalized PageRank, introduced by Jeh and Widom [7] (also see Haveliwala [6]). The personalized PageRank pra(s) depends on two parameters, the jumping constant a and a seed s. A seed can be viewed as a vertex or a probability distribution on vertices. The original definition of PageRank is the special case where the seed is the uniform distribution. To define the PageRank for a connected graph G, we consider random walks on G with transition probability matrix P and the lazy random walk on G, denoted by Z = (I + P)/2. In this paper, all vectors are regarded as row vectors unless stated otherwise. The personalized PageRank vector pra(s) with a jumping constant a and a seed vector s is defined to be the unique solution of the linear system

(1) An alternate but equivalent definition for pr a is an expression of a geometric sum of random walks: 00

pra(s) = as

L (1- alZ

k

.

k=O

In addition to the practical applications of Websearch algorithms, PageRank has numerous connections to various graph invariants. For example, PageRank can be used to find cuts with a certain isoperimetric guarantee, similar to the Cheeger inequalities. It was shown in [1] that for an arbitrarily chosen vertex u, if we arrange vertices in a row using PageRank pra(u), one of the cuts which consists of vertices in initial segments has Cheeger ratio optimal up to a quadratic factor with high probability. The performance guarantee is quite similar to that given by the spectral partitioning algorithm using eigenvectors. However, the advantages of using PageRank are multifold. We can choose an appropriate a to specify the approximate size of the part that we wish to cut and thus PageRank leads to so-called local algorithms. Furthermore, there are effective algorithms for computing

PageRank and Random Walks on Graphs

45

approximate PageRank with finite support with size depending only on the error bound and the desired size of the smaller separated part [1] . In this paper, we explore the relationship of PageRank and various graph invariants occuring in random walks and electrical networks. In the spirit of PageRank, we consider a generalized version of the hitting time and effective resistance with an additional parameter Q . In a way, these generalized invariants provide a quantative ranking of edges, indicating how important an edge is while allowing the choice of Q . In Section 2, we will define the Laplacian and the discrete Green's function as well as their connection to PageRank. In Section 3, we consider electrical networks and the generalized versions of hitting time and the effective resistance. In Section 4, we will give several matrix-forest theorems, which generalize the classical Matrix-Tree Theorem [11]. In Section 5, we will derive a combinatorial interpretation of PageRank in terms of spanning forests in the graph. In Section 6, we consider some useful properties of the generalized hitting time in connection of identifying sparse cuts. In Section 7, we use PageRank to estimate the effective resistance .

2.

LAPLACIAN, THE GREEN'S FUNCTION AND PAGERANK

We consider a connected weighted undirected graph G = (V, E , w). Suppose G has vertex set V, edge set E, edge weight wu,v 2': 0 and IVI = n, lEI = m. A typical random walk is determined by the transition probabilities P(u,v) = wuv/du where the degree d u ofu is the sum L:v :{u,V}EE wu,v ' The volume of a subset 8 ~ V, denoted by vol (8), is the sum of degrees of vertices in 8. In particular, the volume of G, denoted by vol (G), is equal to vol (V) . For the special case of wu,v = 1 for all {u, v} E E , we have vol (G) = 2m. Let A denote the weighted adjacency matrix with entries A( u, v) = wu ,v and D denote the diagonal degree matrix. Then the transition probability matrix P is equal to D- 1 A and for any initial distribution l, the distribution of the random walk after k steps is f pk . For undefined terminology, the reader is referred to the excellent survey of Lovasz [13] The combinatorial Laplacian of G is defined by L = D - A. If we orient the edges of G in an arbitrary but fixed way, we can write its Laplacian as

(2)

46

F. Chung and W. Zhao

where B is the signed edge-vertex incidence matrix, given by

B(e, v) = {

1

if v is e's head

-1

if v is e's tail

a

otherwise

and W is the digonal matrix with W(e, e) = We ' The normalized Laplacian of G is defined to be £ = D- 1/ 2 LD- 1/ 2 and we can write

where S = BD- 1/ 2 . Since E is symmetric and we can express E by n-l

n-l

i= O

i=1

c = 2: )..i a follows from the connectivity of G. The eigenvalue )..1 is intimately related to the rate of convergence of random walks. The reader is referred to [4] for numerous properties concerning eigenvalues of the normalized Laplacian. Although the combinatorial Laplacian is useful for various flow problems in the study of electrical networks, the spectrum of combinatorial Laplacian is not effective (except for almost regular graphs) for applications requiring isoperimetric properties. Denote the ,8-normalized Laplacian £f3 by ,81 + £, . Then we may write Lf3 = SfTWf3S' where we define S' and Wf3 as follows: S'

=

[I] S

B' (n+m)xn

=

[D

1 2 / ]

B

and Wf3 (n+m)xn

=

[,81

a

~]

. (n+m) x (n+m)

For simplicity, we index the columns of S' and the columns of Wf3 by VUE where the first n columns are indexed by V and the last m columns are indexed by E. The rows of Wf3 are indexed in the same way. It is easy to verify that Green's functions were first introduced in a celebrated essay by George Green [8] in 1828. Since then, the concept of Green's functions has been

47

PageRank and Random Walks on Graphs

used in a wide range of areas, especially in the study of partial differential equations and quantum field theory. The discrete analog of Green's functions, which are associated with the normalized Laplacian of graphs , were first introduced in a 2002 paper [5] in connection with the study of Dirichlet eigenvalues with boundary conditions. The Green's function 9 denotes the symmetric matrix satisfying

1£9 = 19£ = f. for all vectors

1 which are orthogonal to the

eigenvector

where 1 denotes the all l 's vector. The Green's function 9 has the following form:

(3) The following modified Green's function 9(3 was also used in [5] . For f3 E jR+ , let Green's function 9(3 denote the symmetric matrix satisfying

(4) Clearly, we have

(5) By comparing with the recurrence of the PageRank in (1), we remark that the discrete Green's function is basically a symmetric form of the PageRank. Namely, we can write

(6) where

(7)

f3=~ . I-a

48

3.

F. Chung and W. Zhao

PAGERANK, THE HITTING TIME AND THE EFFECTIVE RESISTANCE IN ELECTRICAL NETWORKS

A connected weighted undirected graph G = (V, E, w) can be viewed as an electrical network, where the edge weight We represents the conductance of e. The effective resistance R( u, v) between two vertices u and v is defined as the voltage potential difference induced between them when a unit current is injected at u and extracted at v. The effective resistance can be characterized by the combinatorial Laplacian of the graph (see [11]). Since we wish to establish the connection of the effective resistance with PageRank, we will consider the normalized Laplacian instead. Suppose we are given the injected current function iv : V --7 JR. The induced current ie on the edges satisfies the property that the sum of all induced current on edges entering v is equal to iv(v) , as asserted by Kirchoff's current law: iv = iEB.

For any function f : V --7 JR, we can regard f as a voltage potential function in the following sense: The induced flow for the edge from u to v is the product of f(u) - f(v) and the conductance ofthe edge, according to Ohm's law, and can be expressed by

We can write

Suppose we only consider the voltage potential function f satisfying Lv f(v) = O. By using the definition of Green's function , we have

Suppose we inject a unit current to vertex u and extract a unit current from v, i.e., iv = Xv - Xu where Xu is the characteristic function with Xu(x) = 1 if x = u and 0 otherwise. Thus, the effective resistance between vertices u and v can be written as

(8)

R(u, v) =

Ivix» -

Xu?

= ivD- 1/ 2gD- 1/ 2 (Xv -

Xu)T

49

PageRank and Random Walks on Graphs

The effective resistance is closely associated with the hitting time and commute time for a random walk on G. The hitting time H(u, v) is the expected number of steps of a random walk starting from u until it first visit v. The commute time C(u, v) of u and v is the expected number of steps in a random walk starting at u, before vertex v is visited, and the vertex u is reached again. The commute time can be expressed as the sum

C(u, v) = H(u, v) + H(v, u). It has been shown [12] (also see [3]) that the commute time C(u, v) satisfies

R(u,v) C(u, v) = vol (G)" For two distinct vertices u and v, the hitting time H(u, v) satisfies the following equation:

(9)

1

H(u,v) = 1 + d

u

L

H(w,v)

w :{w,u}EE

and H(u, u) = O. Here we will express the hitting time in terms of the Green's function . Lemma 1. For all u, v E V,

H(u, v) _ ( _ )D-1/2gD-1/2 T vol (G) - Xv Xu Xv· Proof. Clearly, H(v, v) = 0 for all v E V. We use the equation in (9). Consider 1 1" -1/2 -1/2 T vol (G) + d L..t (Xv - Xu)D gD Xv u w: (w,u)EE 1 + X D-1/2gD-1/2XT - X D- 1AD-1/2gD-1/2XT v v u v vol (G)

= VOI~G) + XvD-1/2gD-1/2Xr - Xu D- 1/2(I - £)gD- 1/2Xr

50

F. Chung and W. Zhao

= (Xv - Xu)D- 1/2gD-1/2Xr

+ XuD- 1/2 (

1

wl(~

D1/2J D 1/2 + £,g) D- 1/2XT v

= (Xv - Xu)D- 1/2gD-1/2Xr

+ Xu D- 1/2 ( vol1(G) D 1/2J D 1/2 + I

- uT u ) D- 1/2XT 0 0

= (Xv - Xu)D- 1/2gD-1/ 2Xr

+ XuD-1Xr

= (Xv - Xu)D- 1/ 2gD-1/2Xr

•

v

We define

H(u ,v) h(u, v) = vol (G) . Now, we define a generalized hitting time with an additional parameter Q'

> o.

(10) We define the generalized effective resistance Ra(u,v) as follows:

It is easy to check that R a ( u, v) satisfies

where f3 satisfies (7). By using the connection with PageRank in (6), we can write

(11)

Ra (U ,V )

_ [pra(xu)] (u) _ [pra(xu)] (v) -

+

d

u

d;

[pra(xv)] (v) _ [pra(xv)] (u) . dv

du

51

PageRank and Random Walks on Graphs

4.

SEVERAL MATRIX-FoREST THEOREMS

It is well-known that in a simple graph G for an edge joining u and v, the effective resistance R(u, v) is proportional to the number of spanning trees containing the edge {u,v}. Later on we will show that the generalized effective resistance Rex. (u, v) is proportional to a combination of spanning forests of certain types . A forest is a graph containing no cycle. A k-rooted forest is a forest with k connected components where each of the connected components contains a special vertex that we call a root . A k-rooted spanning forest is a k-rooted forest containing all vertices as vertices. It has exactly n - k edges. The special case for k = 1 is a rooted spanning tree. The weight of a forest F is defined to be the product of all weights of edges in F. We define the weight w(F) of an unrooted forest as follows:

w(F) =

II w(e). eEF

For example, if F is a rooted tree in a simple graph, then w(F) = 1. For a rooted tree F, we denote by F* the set of all vertices in F which are not roots . The weight w*(F) of a rooted forest is defined by w*(F) =

ITeEF w(e). ITvEF.

dv

First, we prove several useful facts along the same lines as the MatrixTree Theorem [11] Theorem 1. In a graph G = (V, E, w), for 13 > 0, the determinant of £(3 is related to rooted spanning forests as follows: n

det £(3 =

L 13 L k

k=l

w*(F)

FEFk

where Fk denotes the family of all k-rooted spanning forests in G.

Proof. For a matrix M and subsets X, Y of indices of rows and columns, we denote a submatrix Mx,Y of M by restricting rows and columns of M to X and Y. We consider the determinant of £(3 as follows: det £fj = det SfTWfjS' =

L

(det Sp,v)( det WF,F)(det Sp,v)

FCVUE

IFI=n

52

F. Chung and W. Zhao

by the Cauchy-Binet Theorem. We note that det S~,v = 0 if F contains a cycle. In the other direction, if F does not contain a cycle, F is a spanning forest and the vertices in F n V are the k roots of the spanning forest. Therefore, we have n

det£/3 =

L

L

k=l

F

(detS~,v)2det (W/3)F,F'

k-rooted spanning forest with roots in FnV

Note that for a k-rooted spanning forest F, we have

13 k

detSF V

= ±~

1 d

k

I

,

II .a;

11v d; uEFnv

Thus we have

det Z, =

11

n

Lf3

v v k=l

L

II

F uEFnv k-rooted spanning forests

du

II

We

eEF

n

=

Lf3 L k

k=l

w*(F)

FEFk

as claimed. _ Theorem 2. In a graph G = (V, E, w), for 13 > 0 and a vertex v in G, the determinant of the principle minor £~v), obtained by excluding the row and column associated with v satisfies:

det£~v) =

n

Lf3 k=l

k 1 -

L

w*(F)

FEFk ,v

where Fk,v denotes the family of k-rooted trees having v as one of the roots.

Proof. Let V' denote V \ {v} . We can write the determinant of £~v) as follows:

= FCV'UE IFnv'l=k-l IFI=n-l

53

PageRank and Random Walks on Graphs n

=L k=1

L

(det S~,v,)

2 det (WI') F,F

F k-rooted spanning forest with roots in FnV' and v

since if the forest F satisfies det S~ v #- 0, then F ~ V' U E and F forms a k-rooted spanning forest with v as ~ root . Thus,

and we have

det.C~) =

n

1 d

L

/3k-1

ITu#v u k=1

L

II

F

uEFnv

k-rooted spanning forests v is a root

du

II

We

eEF

n

=

L/3k-1 L

w*(F)

FEFk,v

k=1

as desired. _ Theorem 3. In a graph G = (V, E , w), for /3 > 0 and two distinct vertices u, v in G, the determinant of the minor .c~u,v), obtained by excluding the row associated with u and column associated with v satisfies:

where Fk ,u,v denotes the family of k-rooted spanning forests in which u and v are in the same connected component and u is a root.

Proof. We denote VI = V \ {u} and V2 = V \ {v} . We consider the determinant of .c~u,v) as follows: det .c~u,v) = cr( u , v) det ( (S~lUE,V) T WfJS~2UE,v ) n

=

cr(u,v)

L

L

k=1

FCVUE IFnvl!=k-1 IFI=n-1

(detS~,vJ (detS~,V2) det

(WfJ)F,F

54

F. Chung and W. Zhao n

= u(u,v)

L

L

k=l

FCVUE

(detS~,vJ (detS~,v2) det (Wf3)F,F

IFnV"I=k-l IFI=n-l

since if u or v is in F, one of the determinants in the product of the preceding equality becomes zero. Here u( u, v) denotes (-1) i+j if u is the index of the ith row and v is the index of the jth column of Lf3 . If det S~v is non, 1 zero, the edges in F form a spanning tree with roots {u} U (F n V") where V" = V\ {u, v}. If det S~, v2 is non-zero, then the edges in F form a spanning tree with roots {v} U (F n V") . Therefore if the product is nonzero, vertices u and v are in the same connected component in the spanning forest formed by the edges in F. Suppose F ~ V" U E forms a rooted spanning forest with k connected components, one of which contains both u and v. Thus we have det L~u,v) =

t

u( u, v) J dudv Il#u,v dx k=l

L

FCV"UE IFnv"l=k-l IFI=n-l

We will prove the following Claim later. Claim. det B~,V2 = u(u, v) det B~,Vl' From the Claim, we have det L~u,v) =

1

n

L

J dudv Il#u,v dx k=l

1

n

= J dudv Il#u,v dx L k=l n

F k-rooted spanning forest u is a root u,v are in the same c.c.

L

(det B~,vJ 2 det (Wf3)F,F

FEFk,u,v

L L

II

II

1 {3k-l We dx dudv Ilx#u,v dx k=l FEFk,u ,v eEFnE xEFnv

ITT V

L

55

PageRank and Random Walks on Graphs

=

=

.Jdudv

tj3k-1 k=1

~t

j3k-1

1

k=1

L

IleEFwe

FEFk ,u,v IlxEF.\{v}

L

dx

w*(F).

FEFk ,u,v

It remains to prove the Claim.

Proof of the Claim. We prove the claim in three steps: Case 1: First we see that the Claim holds if F forms a path with vertices U = UI, U2, . . . , Uk+! = v and edges ei = {Ui, Ui+I} in F where Ui is the index for the ith row of L, and ej is the index of the jth column in B . Case 2: Since U and v are in the same connected component of F, let U = VI,V2, . . . ,Vt = V denote the path joining U and V in F. If Ui is the index for the ith row of L, and ej is the index of the jth column in B for 1 ::; i ::; t + 1 and 1 ::; j ::; t, then the claim holds in the same way as in Casel. Case 3: Now we consider the general case. We follow the notation in Case 2 concerning the paths joining U and V in F. We denote by U the permutation on V that moves Vi to the ith place for i = 1, . .. , k + 1 and denote by U' the permutation on E that moves ej = {Vj, vj+d to the jth place for j = 1, ... ,k. We consider

O"(U, v) det (( B~,vJ T B~,V2) = O"(U(u), U(v)) det (UB~,Vl TUfl'U'B'p,V2 V)

We have reduced Case 3 to Case 2. Thus the above expression is equal to 1 and the Claim is proved. The proof of Theorem 3 is completed.

_

56

5.

F. Chung and W. Zhao PAGERANK AND OTHER INVARIANTS IN TERMS OF ROOTED SPANNING FORESTS

Using the matrix-forest theorems in the preceding section, we can establish the connection between the discrete Green's function and rooted spanning forests. We consider the discrete Green's function Q/3 for (3 > O. For a fixed vertex v, we consider

(12) Clearly, f = Q/3Xv is a solution to the above equation. We can view (12) as a linear system with variables f(u) for u in V. By Cramer's rule, we have

= Q/3(v , v) =

det £(v) d et /3

I

(13)

f(v)

(14)

det £~u,v) f(u) = Q/3(U, v) = d £ . et /3

From equation (6) and Theorems 1, 2 and 3, we can express PageRank in terms of the combinatorial sums involving rooted spanning forests as follows:

Theorem 4. For a vertex v in G, the PageRank pra satisfies

where (3 = 2aj (1- a), Fk denotes the family of all k-rooted spanning forests in G and Fk,v denotes the family of k-rooted trees having v as one of the roots.

Theorem 5. For two distinct vertices u and v in G, the PageRank pra satisfies

where (3 = 2aj(1 - a) and Fk,u,v denotes the family of k-rooted spanning forests in which u and v are in the same connected component and u is a root.

57

PageRank and Random Walks on Graphs

As it turns out, the generalized hitting time can also be expressed in a similar form:

Theorem 6. For an edge e = {u, v} in G, the hitting time heA u, v) satisfies

where f3 = 2aj(1 - a) and Fk,e,v denotes the family of k-rooted spanning forests containing e as an edge which is in the path from v to its root.

Proof. From (10), we have

Note that the set of Fk ,v \ Fk,v ,u consists of k-rooted trees in which u and v are in different connected components and v is a root. For every rooted forest F in Fk,v \ Fk,v,u we consider the rooted tree F' formed by adding the edge e = {u, v} where v is no longer a root . The path from v to its root in F' contains e. Clearly, F' has k - 1 connected components and k - 1 roots. Let Fk-l,e,v denote the family of k - 1 forests with the additional property that the path from v to its root contains e. Thus F' is in Fk-l,e,v' We note w*(F') = wew*(F) dv

since v is no longer a root. This is a bijection from Fk ,v \ We can then write

The proof is complete. _

Fk ,v ,u

to

Fk-l ,e ,u-

58

F. Chung and W. Zhao

By using the above theorems, we can then derive a relatively clean formulation for the generalized effective resistance . Theorem 7. For an edge e = {u, v} in G and a E [0, 1), the effective resistance Rex (U, v) satisfies R (

ex

)_

U,

V -

R ",n-l Rk ' "

L..,FEFk,e We L:~=l 13 k L:FEFk

fJ

L..,k=l

fJ

w* (F) w*(F)

where 13 = 2aj(1 - a) and Fk,e denotes the family of k-rooted forests containing e as an edge with k roots.

Proof. The proof follows from the fact that and

Fk,e,u

and

Fk,e,v

are disjoint

Therefore we have

Rex(u, v) = hex(u, v) + hex (v, u)

13 L:~:~ 13 L:FEFk,e w*(F) = k We L:~=l 13 L:FEFk w*(F) . k

6.

•

USING THE GENERALIZED HITTING TIME TO FIND SPARSE CUTS

There are several useful properties of the hitting time hex which we will describe. Lemma 2. For a subset S

c

V with S

L

=1=

hex(u,v)

0, we have

~

{u,v}EE UES,vf!.S

2a l-a

--.

Proof. Since we can view prex(xv) as a probability distribution, we have

L {u,v}EE uES,vf!.S

hex(u,v) =

L {u,v}EE UES,vf!.S

([prex ( :

: )]

(v) _ [prex ( : :

)]

(U))

59

PageRank and Random Walks on Graphs

Since the PageRank vector pra(xv) is the unique solution of the following equation,

I+P

pra(xv) = axv + (1 - a) pra(Xv)-2-' by rearranging the terms above, we have

Thus,

For two distinct vertices, we consider a partition of V into two parts, one of which contains u and the other contains v. In particular, we are interested in the Cheeger ratio of such partitions defined as follows: uv =

min e(8, S) . scv _ min { vol (8), vol (8)}

uES,vES

Theorem 8. For two distinct vertices u and v and a constant E (0,1), suppose two sets X = {w : ha(w,v) ~ o} and Y = {w : ha(w,v) < ha ( u , v)} satisfy the following conditions: .

4a

(1) ha(u,v) > (1- a) vol (X); (ii) vol (X) ~ vol (G)/2 ; (iii) vol (Y)/ vol (X) ~ (1 + /2).

Then uv

~

.

Proof. The proof is by contradiction. We first observe that X c Y. Suppose uv (15)

> . Then we have

e(X, Y) ~ e(X, X) - vol (Y\X)

60

F. Chung and W . Zhao ~

¢>vol (X) - (¢>/2) vol (X)

~

(¢>/2) vol (X) .

by hypothesis and (iii)

By (i) and Lemma 2, we have

I:

ha(x,y) ~

{x ,Y}EE XEX,YEX

I:

ha(x,y)

{x,Y}EE XEX,yEY

since ha(u,v) is the smallest value among all edges (x, y) such that x E X and y E Y. By inequality (15) and (i), the above inequality implies that ~

L.J

2a ha (x , y) > 1 _ a .

{x,Y}EE xEX,yEX

This contradicts Lemma 2 and the proof is complete. _

7. USING PAGERANK TO ESTIMATE THE EFFECTIVE RESISTANCE

The effective resistance R(u, v) is a very useful graph invariant. For example, the recent work by Spielman and Srivastava [14] on graph sparsification relies on the effective resistance . Here we wish to illustrate that we can use the generalized effective resistance to approximate the effective resistance. Then by using the effective approximation algorithm for computing PageRank [1] and the PageRank representation in (11) for the generalized effective resistance, we can approximate the effective resistance as a result. Recall that for a = f3/(2 + (3) , we have

Ra(u, v) = f3(Xu - Xv)D- I / 2Y/3D- 1/ 2(Xu - Xvf · Theorem 9. For two distinct vertices u and v in G, we have

If3R(u, v) -

Ra(u, v)1

::; f32Ai

( 1

1)

d +d u v

where Al is the smallest nontrivial eigenvalue of L of G and a = f3 / (2 +(3) .

61

PageRank and Random Walks on Graphs

Proof. From (3) and (8), we have

R(u, v) = (Xv - Xu)D- 1/2gD- 1/2(Xv - Xu)T =

~ ~ (
_
~

i=l Ai

.a;

From equation (5), we have

Ra(u,v) = ~ _f3_ ( 0,

If3R(U,v) - Ra(u,v)1 :::; (
(tAi __Ai f3+ f3 ) (
~

f32 (
2

since
Thus we have

If3R(u ,v) <

f32 - A1(A1 + (3) f32 :::; Ai

Ra(u,v)\

(2 ((
(2

1

vol (G) + du

-

1

< f32 (~+~) - Ai

since

I«

du

1

J du vol (G) + dv

d;

and fv are orthonormal vectors. -

-

1)

J d; vol (G)

62

F . Chung and W. Zhao

REFERENCES [1] R. Andersen, F. Chung and K. Lang , Local graph partitioning using pagerank vectors, Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (2006), pp. 475-486.

[2] S. Brin and L. Page , The anatomy of a large-scale hypertextual Web search engine, Computer Networks and ISDN Systems, 30(1-7) (1998), pp. 107-117 .

[3] A. k. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky and P. Tiwari , The electrical resistance of a graph captures its commute time and cover time, Proceedings of the 26th ACM Symposium on Theory of Computing (1989), pp . 574-586 . [4]

F. Chung , Spectal Graph Theory, AMS Publication, 1997.

[5] F. Chung and S.-T . Yau, Discrete Green 's Functions. Journal of Combinatorial Theory, Series A , 91(1-2) (2000), pp. 191-214.

[6] H. Haveliwala, Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search , IEEE Trans. Knowl. Data Eng., 15(4) (2003), pp . 784-796.

[7] Glen Jeh and Jennifer Widom, Scaling personalized web search. Proceedings of the 12th World Wide Web Conference (WWW) (2003), pp. 271-279 .

[8] G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828.

[9] S. Guattery and G. L. Miller, Graph embeddings and laplacian eigenvalues, Journal on Matrix Analysis and Applications, 21(3) (2000), pp. 703-723. [10]

G. Kirchhoff, Uber die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefiihrt wird, Ann. Phys . chem. , 72 (1847), 497-508.

[11]

G. Kirchhoff, On the motion of electricity in conductors (Uber die Bewegung der Elecktricitiit in Leitem), Annalen der Physik, Vol. 102 (1857), pp . 529.

[12]

C. St . J . A. Nash-Williams, Random walks and electronic currents in networks , Proceedings of the Cambridge Philosophical Society , 55 (1959), pp. 181-194.

[13]

L. Lovasz, Random walks on graphs: A survey, Combinatorics, Paul Erdos is Eighty, 2 (1993), pp . 1-46.

[14]

D. A. Spielman and N. Srivastava, Graph sparsification by effective resistances, Proceedings of the 40th ACM Symposium on Theory of Computing (2008), pp . 563-568.

Fan Chung and Wenbo Zhao University of California, San Diego, La Jolla, CA 92093, US e-mail:

{fan.w3zhao}~ucsd.edu

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 63-93.

SOLUTION OF PETER WINKLER'S PIZZA PROBLEM*t

JOSEF CIBULKA, JAN KYNCL, VIOLA MESzAROS, RUDOLF STOLAR and PAVEL VALTR

Bob cuts a pizza into slices of not necessarily equal size and shares it with Alice by alternately taking turns. One slice is taken in each turn. The first turn is Alice's. She may choose any of the slices. In all other turns only those slices can be chosen that have a neighbor slice already eaten. We prove a conjecture of Peter Winkler by showing that Alice has a strategy for obt aining 4/9 of the pizza. This is best possible, that is, there is a cutting and a strategy for Bob to get 5/9 of the pizza. We also give a characterization of Alice's best possible gain depending on the number of slices. For a given cutting of the pizza, we describe a linear time algorithm that computes Alice's strategy gaining at least 4/9 of the pizza and another algorithm that computes the optimal strategy for both players in any possible position of the game in quadratic time . We distinguish two types of turns, shifts and jumps. We prove that Alice can gain 4/9, 7/16 and 1/3 of the pizza if she is allowed to make at most two jumps, at most one jump and no jump, respectively, and the three constants are the best possible.

1.

INTRODUCTION

Peter Winkler posed the following problem at the conference Building Bridges, honouring the 60th birthday of Laszlo Lovasz, in Budapest in 2008. Bob and Alice are sharing a pizza. Bob cuts the pizza into slices "Work on this paper was supported by the projects 1M0545 and MSM0021620838 of the Ministry of Education of the Czech Republic . Viola Meszaros was also partially supported by OTKA Grant K76099. Josef Cibulka and Rudolf Stolar were also supported by the Czech Science Foundation under the contract no. 201/09/H057. t A short version of the paper appeared in Combinatorial Algorithms, the proceedings of the IWOCA 2009, Hradec nad Moravici, Czech Republic , LNCS, Vol. 5874, pp . 356-368 , Springer (2009) .

64

J. Cibulka, J. Kync], V. Meszaros , R. Stolar and P. Valtr

of not necessarily equal size. Afterwards they take turns alternately to divide it among themselves. One slice is taken in each turn. In the first turn Alice takes any slice. In the forthcoming turns one may take a slice if it is adjacent to some previously taken slice. This is called the Polite Pizza Protocol. How much of the pizza can Alice gain? The original puzzle, that is determining whether Bob can get more than half of the pizza, was devised by Dan Brown in 1996. Bob can easily ensure for himself one half of the pizza. For example, he may cut the pizza into an even number of slices of equal size. Then Bob always obtains exactly one half. Peter Winkler found out that Bob can actually get 5/9 of the pizza if he cuts the pizza properly - see Theorems 4 and 5 for such cuttings. He conjectured that Alice can obtain 4/9 of the pizza for any cutting. The main aim of this paper is to show a strategy of Alice proving this conjecture. The pizza after Bob's cutting may be represented by a circular sequence P = POPl .. . Pn-l and by the sizes IPil ~ 0 (for i = 0,1 , .. . , n - 1); for simplicity of notation, throughout the paper we do not separate the elements of (circular) sequences by commas. The size of P is defined by !PI := L:~:ollpil· Throughout the paper the indices are counted modulo n. For 1 < j ::; n , if one of the players chooses a slice Pi in the (j - 1)-st turn and the other player chooses Pi-lor Pi+! in the j-th turn, then the j-th turn is called a shift, otherwise it is called a jump. Except for the first and the last turn, there are two choices in each turn and exactly one of them is a shift and the other one is a jump. The last turn is always a shift. If some strategy of a player allows the player to make at most j jumps, then we call it a j -jump strategy. We remark that given a circular sequence P of length n, Alice has exactly n zero-jump strategies on P, determined by Alice's first turn. Let E be a particular strategy of one of the players. We say that E is a strategy with gain 9 if it guarantees the player a subset of slices with the sum of sizes at least g. Note that according to this definition, if E is a strategy with gain 9 then it is also a strategy with gain g' for any g' ::; g. If the number of slices is even, Alice has the following zero-jump strategy with gain IPI/2. She partitions the slices of the pizza into two classes, even and odd, according to their parity in P. In the first turn Alice takes a slice from the class with the sum of slice sizes at least IP\/2. In all her forthcoming turns she makes shifts, thus forcing Bob to eat from the other class in each of his turns.

Here is our main result.

Solution of Peter Winkler's Pizza Problem

65

Theorem 1. For any P, Alice has a two-jump strategy with gain 4IPI/9. More generally, we determine Alice's guaranteed gain for any given number of slices.

Theorem 2. For n ~ 1, let g(n) be the maximum 9 E [0,1] such that for any cutting of the pizza into n slices, Alice has a strategy with gain giPI. Then if n=l, I

g(n) =

4/9 if n E {15, 17, 19, ... }, {

1/2

otherwise.

Moreover, Alice has a zero-jump strategy with gain g(n)IPI when n is even or n ::; 7, she has a one-jump strategy with gain g(n)IPI for n E {9, 11, 13}, and she has a two-jump strategy with gain g(n)IPI for n E {15,17 ,19, . . . }. If we make a restriction on the number of Alice's jumps we get the following results.

Theorem 3. (a) Alice has a zero-jump strategy with gain 1P1/3 and the constant 1/3 is the best possible. (b) Alice has a one-jump strategy with gain 71PI/16 and the constant 7/16 is the best possible. Due to Theorem 2, the following theorem describes all minimal cuttings for which Bob has a strategy with gain 5IPI/9.

Theorem 4. For any w E [0,1], Bob has a one-jump strategy with gain 51PI/9 if he cuts the pizza into 15 slices as follows: Pw = 0010100(1+w )0(2w)00202. These cuttings describe, up to scaling, rotating and flipping the pizza upside-down, all the pizza cuttings into 15 slices for which Bob has a strategy with gain 5IPI/9. For w = 0 or w = 1, the cutting in Theorem 4 has slices of only three different sizes 0,1,2. If all the slices have the same size, then Alice always gets at least half of the pizza. But two different slice sizes are already enough to obtain a cutting with which Bob gets 5/9 of the pizza.

Theorem 5. Up to scaling, rotating and flipping the pizza upside-down, there is a unique pizza cutting into 21 slices of at most two different sizes for which Bob has a strategy with gain 5IPI/9. The cutting is 001010010101001010101.

66

J. Cibulka, J. Kyucl, V. Meszaros, R. Stolar and P. Valtr

In Subsection 7.1 we describe a linear-time algorithm for finding Alice's two-jump strategy with gain g(n)IPI guaranteed by Theorem 2. Theorem 6. There is an algorithm that, given a cutting of the pizza with n slices, performs a precomputation in time O(n) . Then, during the game, the algorithm decides each of Alice's turns in time 0(1) in such a way that Alice makes at most two jumps and her gain is at least g(n)IPI. There is also a straightforward quadratic-time dynamic algorithm finding optimal strategies for each of the two players.

Claim 7. There is an algorithm that, given a cutting of the pizza with n slices, computes an optimal strategy for each of the two players in time 0(n 2 ) . The algorithm stores an optimal turn of the player on turn for all the n 2 - n + 2 possible positions of the game. We remark that, unlike in Theorem 1, the number of Alice's jumps in her optimal strategy cannot be bounded by a constant. In fact, it can be as large as ln/2J-1 for n ~ 2 (see Observation 28 in Subsection 7.2). A similar statement holds for the number of Bob's jumps in his optimal strategy. The following question is still open .

Problem 1. Is there an algorithm that uses o(n 2 ) time for some precomputations and then computes each optimal turn in constant time? We remark that we even don't know if Alice's optimal first turn can be computed in time o(n 2 ) . Independently of us and approximately at the same time, K. Knauer, P. Micek and T . Ueckerdt [3] also proved Theorem 1 and some related results. The paper is organized as follows. Theorem 1 is proved in Section 2. Section 3 contains examples of cuttings showing that the constant 4/9 in Theorem 1 cannot be improved. Section 4 is devoted to the proof of Theorem 2. Theorems 4 and 5 are proved in Section 5. Section 6 contains the proof of Theorem 3. The algorithms from Theorem 6 and Claim 7 are described in Subsection 7.1 and Subsection 7.2, respectively.

In a follow-up paper [1] we discuss generalizations of the Pizza Problem.

67

Solution of Peter Winkler 's Pizza Problem

2. THE LOWER BOUND

1P1/2.

When the number of slices is even, Alice can always gain at least Here we prove the lower bound on her gain when n ;::: 3 is odd.

2.1. Preliminaries If the number of slices is odd, instead of the circular sequence P = POPl " ·Pn-l we will be working with the related circular sequence V = VOVl . .. Vn-l = POP2·.· Pn -lPlP3 ... Pn-2 that we call the characteristic cycle (see Figure 1). The size of the characteristic cycle is denoted by IVI. Clearly IVI = IPI· Vo = Po Vn-I

=

Pn-2

•

•

VI

•

• • •

v~ =PI 2

•2

Vn 1

=

P2

• • =

Pn- I

Fig. 1. A cutting of a pizza and the corresponding characteristic cycle

An arc is a sequence of at most n - 1 consecutive elements of V. If we talk about the first or the last element of an are, we always consider it with respect to the linear order on the arc inherited from the characteristic cycle V. For an arc X = ViVi+1 . . . Vi+l-l, its length is l(X) := l and its size is IXI := L:~~li-llvjl. An arc of length (n + 1)/2 is called a half-circle . Figure 2 shows an example of a game on V . The slice taken in the i-th turn is labelled by the initial letter of the player with i in the subscript. At any time during a game, a player may decide to make only shifts further on. The player will take one or two arcs of the characteristic cycle afterwards. An example of such a game when Alice decided to make no more jumps is depicted on Figure 3 (slices taken before the decision point are labelled with *, and selected pairs of slices neighboring in the original pizza are connected by dashed segments) . The slices she took after the

68

J. Cibulka, J. Kyncl , V. Meszaros, R. Stolar and P. Valtr

A3

B

- -

_

6

Al

-A7

B4-

-

B2

- A5

Fig. 2. A game illustrated on the characteristic cycle V (the turns are AI, B2, A 3 , . •. ). The turns B 4 and A 5 are jumps and all the other turns (except AI) are shifts

decision point are forming two arcs that are separated in between by some arc of previously taken slices.

•

-

B I 4-

- - - --

• Fig. 3. Situation before Bobs turn with the two possible options marked by arrows (left) and two of the possible ends of the game where Alice made no more jumps (middle and right)

Observation 8. Consider a position after Alice's turn Aj,j =1= 1, n. We have V = TIRIT2R2 , where £(TI) = £(T2) + 1 = (j + 1)/2, £(R I) = £(R 2), T I and T 2 are two arcs of already taken slices, and R I and R2 are two arcs containing the remaining slices. Suppose that all the remaining turns of Alice (Aj+2' Aj+4, "" An) are shifts. Then, regardless of Bob's remaining turns B j +1 , " " Bn-l, the slices taken by Alice in the turns Aj+2' Aj+4,"" An necessarily form two arcs Xl and X2 such that XITIX2 is a half-circle of V. In addition, for any half-circle YITIY2, Bob can choose his turns Bj+2' ... ,Bn- l so that Xl = YI and X 2 = Y2.

Proof. We will show by induction that before any Bob's turn Bj+2k+b the slices taken by him in turns B j +1 , " " B j+2k-1 form two arcs Zl and Z2 such that Z = ZlT2Z2 is an arc and his two possible moves are on the

69

Solution of Peter Winkler's Pizza Problem

two neighbors of Z. This is true for B j+1 and by induction if this is true before Bj+2k+1' then Bob takes for Bj+2k+1 one of the two neighbors of Z and Z' := Z U Bj+2k+1 is an arc. After Alice's shift, Bob's shift would be a neighbor of Bj+2k+1' thus a neighbor of Z'. Bob's jump would be the neighbor of Z different from B j +2k+1' thus a neighbor of Z' . For any given half-circle YITIY2 and before any of Bob's turns Bj+l,"" B n - l , the two slices available for Bob are neighbors of an arc of length at most (n - 3)/2 which is not a subarc of Y ITIY2 • Thus one of the two slices available for him is not in YITI l'2 and Bob can choose his turns Bj+1," " B n - l so that Xl = Y I and X2 = Y2. •

*~ITl,r*

•

*• •* -

*•

•

III I,'" I} ,

•

II 1/

~

I

B w• R 2

'I ,

•

,

,

•

1\ 1'1 1/

~12.

* Fig .

I

h,.

•

B 14• B 1;

•

•4

B

T2

*'

•

•

Ag

A7

As

4. Two possible choices of Bob's next

turn (left) and the two possible ends of the game where Bob made no more jumps (middle and right)

If Bob decides to make only shifts for the rest of the game, he takes one arc afterwards. Namely, if there are two arcs of already taken slices in V at his decision point, then the arc that will be taken by Bob is neighboring these two arcs at both of its ends (see Figure 4). Observation 9. Consider a position after Alice's turn Aj,j i= 1,n. We have V = TIRIT2R2, where f(TI) = f(T2) + 1 = (j + 1)/2, f(RI) = f(R 2), TI and T2 are two arcs of already taken slices, and RI and R2 are two arcs containing the remaining slices. Bob's turn B j+1 may be on the last slice of R I or on the first slice of R2. If B j+1 is on the last slice of RI and all the remaining turns of Bob are shifts then , regardless of Alice's remaining turns, Bob will take RI and Alice will take R2 in this phase of the game. Similarly, if B j+1 is on the first slice of R2 and all the remaining turns of Bob are shifts then, regardless of Alice's remaining turns, Bob will take R2 and Alice will take R I in this phase of the game. Proof. Similarly to the proof of Observation 8, it is easy to prove by induction that if Bob played Bj+1 on R I, then before each Alice's turn, the two slices available for her are from R2. •

70

J. Cibulka., J. Kyticl, V. Meszaros, R. Stolar and P. Valtr

2.2. Minimal triples For each v in V the potential of v is the minimum of the sizes of half-circles covering v. The maximum ofthe potentials in V is the potential of V, which we further denote by p(V) . It is an immediate conclusion that Alice has a strategy with gain p(V) because by choosing an element with potential equal to p(V) and making only shifts afterwards Alice obtains at least p(V) . Therefore we may assume that p(V) < 1V1/2. A covering triple of half-circles is a triple of half-circles such that each element of V appears in at least one of the three half-circles . We allow two half-cirIes to be equal in the covering triple. A covering triple is minimal if it contains a half-circle of minimum size (among all n half-circles), all halfcircles forming the triple have size at most p(V) and none of them may be replaced in the triple by a half-circle of strictly smaller size.

Claim 10. Each half-circle of minimum size lies in at least one minimal triple. Proof. Take a half-circle HI of minimum size. Consider Vk and vk+(n-3)/2 the two uncovered elements neighboring HI on V . Let H 2 be the halfcircle of size at most p(V) that covers Vk and as many elements of V not covered by HI as possible. We define H 3 in the same way for Vk+(n -3) /2' The above triple of half-circles covers V. If it is not the case, then take an uncovered element v . Consider a half-circle H that has minimal size among half-circles covering v. At least one of Vk and vk+(n-3)/2 is covered by H. This contradicts the choice of H2 or H 3 . So we get that the given triple of half-circles forms a covering triple. Now while any of the half-circles can be replaced in the triple by a half-circle of strictly smaller size, we replace it . Obviously HI won't be replaced as it is a half-circle of minimum size. Consequently the triple we get is a minimal triple. _ Observation 11. If the size of a half-circle in a minimal triple is z then Alice has a zero-jump strategy with gain z. Proof. As in a minimal triple all half-circles are of size at most p(V) and Alice has a zero-jump strategy with gain p(V), the statement of the observation follows. _ Claim 12. Let p(V) < 1V1/2. Then any minimal triple contains three pairwise different half-circles, and thus there is a partition of V into six

Solution of Peter Winkler's Pizza Problem

71

arcs A, B, C, D, E, F such that the half-circles in the minimal triple are ABC, CDE and EFA (see Figure 5). The lengths of the arcs satisfy l(A) = l(D) + 1 ~ 2, l(C) = l(F) + 1 ~ 2 and l(E) = l(B) + 1 ~ 2.

Fig . 5. The partitioning of the characteristic cycle given by the covering half-circles

Proof. If two of the three half-circles in a minimal triple are equal then V can be covered by two half-circles of the triple. Since each half-circle in the triple has size at most p(V), the total size of the pizza is at most 2p(V) < lVI, a contradiction. If at least one of B, D, F has length 0, we argue exactly in the same way.

We have l(ABC) + l(EFA) = n + 1 = l(A) + ...+ l(F) + 1, therefore l(A) = l(D) + 1 ~ 2. The other two equalities are analogous. _

2.3. An auxiliary one-jump strategy

Throughout this section we assume that p(V) < 1V1/2. We fix any minimal triple T of half-circles. By Claim 12, it yields a partition of V into six arcs A, B , C, D, E, F such that the half-circles in the triple are ABC, CDE, EFA (see Figure 5). We further use the notation a := IAI, b := IBI, and so on. We define a median slice of an arc X = ViVi+l ' " Vi+l to be a slice Vk EX such that IVjl ~ IXI/2 and I:,~~~+llvjl ~ IXI/2. Observe that any arc of positive length has at least one median slice.

I:,J:f

Claim 13. Alice has a one-jump strategy for V with gain b/2 + min {c + d,

f + a} if p(V) < /V1/2.

72

J. Cibulks , J. Kyn cl, V. Meszaros, R. Stolar and P. Valtr

Proof. By Claim 12 we have that l(B) > O. In the first turn Alice takes a median slice Vk of B. Consequently Bob is forced to start in E. He may t ake the element Vk+( n -l )/2 or v k+ (n+1)/2' Alice makes only shifts while the shift implies taking an element of B . In the meantime Bob necessarily takes elements from E . In the turn, when Alice's shift would imply t aking an element outside of B , Alice makes a jump instead. In that moment some init ial arc Eo of E st arting from the boundary of E is already taken. Let El be th e remaining part (subarc) of E . Alice takes the available element of E 1 . Note that such an element exists since in the sequence P , all the neighbors of the slices of B are in E. (see Figure 6). The half-circle X 1EoX2 can replace either CDE or EFA in the fixed minimal triple. Thus due to the minimality of the triple, the size of X 1X2 is always at least the size of either CD or FA. As Alice obtained at least the half of B before the jump, in the end she gains at least b/2 + min {c + d, f + a}. •

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Fig. 6. One-jump st rategy: Alice chooses a jump rather than a shift (left) and makes no

more jumps afterwards (right )

Corollary 14 . Alice has a one-jump strategy for V with gain (a+b+ c)/4+ (d + e + 1)/2 if p(V) < 1V1/2. Proof. By Claim 13 Alice has a strateg y with gain b/2 + min {c + d, f + a}. Without loss of generality we may assume thi s sum is gl := b/2 + c + d. Alice also has a strat egy with gain g2 := e + f + a by Observation 11. Combining the two results Alice has a gain max {gl ' g2} 2: g1/2 + g2/2 = (a + c + d + e + 1)/2 + b/4 2: (a + b + c)/4 + (d + e + 1)/2. •

Solution of Peter Winkler's Pizza Problem

73

2.4. A two-jump strategy Throughout this subsection we assume that p(V) < 1V1/2 and that V is partitioned into six arcs A, ... , F in the same way as in the previous subsection. In this subsection we describe a strategy satisfying the following claim.

Claim 15. Alice has a two-jump strategy for V with gain b/2 min {c + d, f + a} if p(V) < 1V1/2.

+ e/4 +

2.4.1. Two phases of the game. Let B = ViVi+l . . . Vi+~ ' Then E = VjVj+l .. . vj+~+}, where j = i + (n - 1)/2. Consider the circular sequence V' = ViVi+l .. . Vi+~ Vj Vj+l . . . Vj+~+l obtained by concatenating the arcs B and E . Let H be a half-circle of V' containing Vj' Then its size is not smaller than the size of E, since otherwise the half-circle CD E of V could be replaced in the minimal triple T by a half-circle of smaller size - namely by the half-circle formed by the slices contained in CD and in H . Similarly, if H is a half-circle of V' containing vj+~+l, then its size is also not smaller than the size of E. Since each half-circle of V' contains "i or vj+~+}, it follows that E is a half-circle of V' of minimum size. If p(V') 2: 1V'1/2 then Alice has a zero-jump strategy E for V' with gain p(V') 2:: 1V'1/2 2:: b/2 + e/4. Otherwise, by Corollary 14 (applied on V'), Alice has a one-jump strategy E for V' with gain b/2 + e/4 (we use the fact that E is a half-circle of V' of minimum size, and therefore it is contained in a minimal triple yielding a partition of V' into six arcs A' , B' , ... ,F' such that E = A' B'G' and B = D' E' F') . Briefly speaking, Alice's strategy on V follows the strategy E as long as it is possible, then Alice makes one jump and after that she makes only shifts till the end of the game. In the rest of this subsection (Subsection 2.4), we consider a game G on V . We divide the turns of G into two phases. The first phase of G is the phase when Alice follows the strategy E and it ends with Bob's turn. Alice's first turn that does not follow (and actually cannot follow) the strategy E is the first turn of the second phase of G. It is always a jump and all the other turns of Alice in the second phase are shifts . We now describe Alice's strategy in each of the two phases of G in detail.

74

J . Cibulka, J . Kyncl , V. Meszaros, R. Stolar and P. Valtr

2.4.2. Alice's strategy in the first phase. As mentioned above, Alice has a one-jump strategy E for V' with gain b/2 + e/4. We now distinguish two cases.

Case 1: The strategy E is a zero-jump strategy. Let the first turn in the zero-jump strategy E be on a slice q E V'. The first turn could be also on any other point of V' with the same or larger potential. Observe that the potentials of the slices in V' are e on E and at least e on B. Therefore we may assume that q lies in B . In the game G, Alice makes her first turn also on q. In the second turn Bob can choose between two slices in E . In the subsequent turns Alice makes shifts as long as Bob's previous turn was neither on the first nor on the last slice of E. Consider all slices taken by Bob up to any fixed moment during the first phase of the game G. They always form a subarc of E (and the slices taken by Alice form a subarc of B). The first turn in which Bob takes the first or the last slice of E is the last turn of the first phase. Note that after that Alice's shift would be either on the last slice of A or on the first slice of C (see Figure 7). But Alice makes a jump and this jump is the first turn of the second phase. Note that this jump is in E (see Figure 7).

• •

• • • D •

.-:

Fig . 7. Case 1: After the end of first phase, Alice chooses a jump rather than a shift

(two examples shown)

Case 2: The strategy E is not a zero-jump strategy. Following the proof of Corollary 14, we may suppose that E is the strategy that we describe below. By Claim 10, the half-circle E of minimum size is contained in some minimal triple T' of half-circles of V'. The triple T' determines a partition

75

Solution of Peter Winkler's Pizza Problem

of V' into six arcs A' , B' , ... , P' in the same way as T determined a partition of V into A, B , . .. , F. We may suppose that E = A' B' 0' and B = D' E'P' . We may suppose that the size of B' is positive, since otherwise one of the half-circles 0' D' E' and E'P' A' has size at least b/2+ e/2 and thus Alice has a zero-jump strategy for V' with gain b/2 + e/ 2, allowing us to use the above Case 1. In the first turn Alice takes a median slice of B'. Then in the second turn Bob can choose between two slices of E'. In the subsequent turns Alice makes shifts as long as Bob's previous turn was neither on the first nor on the last slice of E'. In each moment in this part of the game Bob's turns form a subarc of E' . At the first instance when Bob takes the first or the last slice of E' , Alice makes a jump, which is always in E' (see Figure 8). Note that so far the game was an analogue of the first phase in Case 1, with B' and E' in place of B and E, respectively. After her first jump Alice makes shifts as long as Bob's previous turn was neither on the first nor on the last slice of E. Note that Bob's turns in this part of the game are in E (see Figure 8). At the first instance when Bob takes the first or the last slice of E , Alice makes a jump, which is already the first turn of the second phase. This jump is necessarily in E (see Figure 8).

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then she makes an other jump after the end of th e first phase (right)

76

J. Cibulka, J. Kyncl, V. Meszaros, R. Stolar and P. Valtr

2.4.3. Alice's strategy in the second phase. Alice's strategy in the second phase is very simple. Above we describe the first phase and also the first turn of the second phase, which is always a jump done by Alice. In the rest of the second phase Alice makes only shifts. 2.4.4. Analysis of Alice's gain. Since the first phase of G ends by Bob's turn on the first or on the last slice of E, we may suppose without loss of generality that it ends with Bob's turn on Vj ' Then the part of V removed in the first phase of G is a union of some initial subarc Bo of B and some initial subarc Eo of E. Let E I be the arc formed by the slices of E not taken in the first phase of G, thus E = EoE I , and letel := JEll. In her jump at the beginning of the second phase of G, Alice takes the first slice of E I . By Observation 8, all the slices taken by Alice in the second phase of G form two arcs Xl and X 2 such that X IEoX 2 is a half-circle of V (see Figure 6). Since none of the half-circles CDE and ·EF A can be replaced in the triple T by a half-circle of a strictly smaller size, the sum IXII + IX2 1 achieves its minimum either for Xl = CD and X 2 = E I , or for l(X I ) = 0 and X2 = ElF A . Thus, the portion collected by Alice in the second phase of G is at least ei + min {c + d, f + a}. Now, consider an auxiliary game G' on V' consisting of two phases defined as follows. The turns in the first phase of G' are exactly the same as the turns in the first phase of G (this is a correct definition, as all turns in the first phase of G are in B U E and the first or the last slice of E is taken only at the very end of the first phase). In the second phase of G', both Alice and Bob make only shifts. We claim that Alice actually follows the one-jump strategy E in the whole game G'. This is obvious in the first phase of G'. Further, if E is a zero-jump strategy then Alice clearly follows E also in the second phase of the game G'. Otherwise Alice makes her only jump in the first phase of the game (see Case 2 in Paragraph 2.4.2) and thus again she follows E also in the second phase of G'. By Observation 9, Alice collects exactly the slices of E I in the second phase of G'. Thus , if 9 denotes the portion collected by Alice in the first phase of G' then 9 + el is her portion in the whole game G'. Since the strategy E guarantees gain b/2 + e/4 to Alice, we get 9 + el ~ b/2 + e/4. Alice's portion in the whole game G is at least 9

+ (el + min {c + d,

f + a}) 2: b/2 + e/4 + min {c + d, f + a}, which completes the proof of Claim 15.

77

Solution of Peter Winkler's Pizza Problem

2.5. Proof of the lower bound Proof of Theorem 1. If the number of slices is even then Alice has a zero-jump strategy with gain 1P1/2. We further suppose that the number of slices is odd. We consider the characteristic circle V. If p(V) 2': 1V1/2 then Alice has a zero-jump strategy with gain 1V1/2 = IPI/2. Suppose now that p(V) < 1V1/2. Then V may be partitioned into six arcs A, . . . , F as in Claim 12. Without loss of generality, we may assume that a+b+c:S; c+d+e:S; e+ f +a. Thus, a+b:S; d+e and c+d:S; f +a. By Observation 11, Alice has a zero-jump strategy with gain

91 := e + f + a. By Claim 15, Alice has a two-jump strategy with gain

92 := b/2 + e/4 + min {c + d, f

+ a} =

b/2 + e/4 + c + d.

By an analogue of Claim 15, Alice also has a two-jump strategy with gain

93

:=

f /2 + c/4 + min {a + b, d + e} = J/2 + c/4 + a + b.

One of the three strategies gives gain

max {91 ,92 ,93} 2': (391

+ 492 + 293)/9

= (5a + 4b + 9c/2 + 4d + 4e + 4J)/9 = (41PI + a + c/ 2) /92': 4IPI/9.

3.

•

THE UPPER BOUND

In this section we show a strategy for Bob that guarantees him 5/9 of the pizza. Then Bob has to cut the pizza into an odd number of slices, since otherwise Alice has a strategy with gain IP!/2, as was observed in the introduction. Before each turn of Bob, the number of the remaining slices is even. The sequence of all the remaining slices can be then written as PiPi+lPi+2· .. Pi+2j -l·

Let K := PiPi+2 .. . Pi+2j-2 and L := Pi+1Pi+3 . . . Pi+2 j-l be the sequences of slices on odd and even positions respectively. We use the following reformulation of Observation 9:

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J. Cibulka, J. KynCl, V. Meszaros, R. Stolar and P. Valtr

Observation 16. Before any of his turns, Bob has a strategy that guarantees him max {IKI, ILl} in addition to what he already has. In the strategy Bob makes only shifts, except possibly for the first turn. Proof. We prove that Bob can get all slices from L . A similar proof shows that he can get all slices from K . In his first turn, Bob takes Pi+2j-l E L. In each other turn, Bob makes shifts. Then before each of Alice's turns, the two slices available for her are from K. •

Claim 17. Bob has a one-jump strategy with gain 51PI/9 if he cuts the pizza into 15 slices in the following way: 002020030300404. Proof. The size of the pizza is 18, which means that Bob wants to get slices with sum of sizes at least 10. We consider all possible first moves of Alice: 1. Alice takes a zero slice located between two nonzero slices. The sizes of the slices remaining after her turn are QOQbQbQOfOfOQa,

where the elements of K are underlined and a, band c are in one of the six possible bijections with 2, 3 and 4. Then IKI = 2c + a and ILl = 2b + a and by a case analysis of the possible values of a, band c, Bob gets max { IKI, ILl} ~ 10. 2. Alice takes a zero slice located between a zero slice and a nonzero slice. This leads to QOQOQbQbQOfOfO and Bob gets max {IKI, ILl} = max {2a

+ 2c, 2b} = 2a + 2c ~ 10.

3. Alice takes a nonzero slice. The situation is then QaQOQOQOQcQcQO.

Bob now takes the rightmost slice and then makes shifts until he either takes a or the two slices c. This leads to three possible cases:

(a) QOQOQOQcQcQ,

(b) QOQOQOQcQ,

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Solution of Peter Winkler's Pizza Problem

(c) QaQOQOQOQ.

After Alice takes one of the available zero slices, we use Observation 16 to show that the gain of Bob in these three cases is (a) a + max {IKI, ILl} = a + max {2b, 2c} , (b) a + c+ max { IKI , ILl} (c) 2c + max {IKI, ILl}

= a + c+ max{2b,c} = a + c+ 2b,

= 2c + max {2b, a} = 2c +

2b.

In any of the three cases and for any bijective assignment of the values 2, 3 and 4 to a, band c, Bob gets slices of total size at least 10. •

Corollary 18. For any w E [0,1], Bob has a one-jump strategy with gain 51PI/9 if he cuts the pizza into 15 slices as follows: Pw = 0010100(1 +w)0(2w)00202. Proof. The claim holds for w = 1/2, since P1 / 2 is a scale-down of the pizza considered in Claim 17. Clearly, if some slices of P1 / 2 have total size at least 5 then also the corresponding slices of Pw , w E [0,1], have total size at least 5. Therefore, Bob can ensure gain 51Pwl/9 = 5 for Pw , w E [0,1], with the same strategy as for P1 / 2 • • In Section 4 we show that Bob has no strategy with gain 51PI/9 for pizza cuttings with fewer than 15 slices. Moreover, in Section 5 we show that Corollary 18 describes essentially all cuttings into 15 slices that guarantee Bob 5/9 of the pizza .

Claim 19. For any odd n ~ 15, Bob has a one-jump strategy with gain 51PI/9 using some cutting of the pizza into n slices. Proof. For n = 15 the claim follows from Claim 17. For larger n, we take the cutting P from Claim 17 and add n - 15 zero slices between some two consecutive zero slices in the sequence . If Alice starts in one of the added slices, then the situation is similar to the case 2 of the proof of Claim 17. The only difference is that there might be additional zeros at the beginning and at the end of the sequence . But these zeros either do not change the values of IKI and ILl or swap the two values. Thus Bob can get max {IKI ,ILl} = 10.

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J. Cibulka., J. Kyucl, V. Meszaros, R. Stolar and P. Valtr

Otherwise Bob uses the strategy from the proof of Claim 17. In cases 1, 2, the even number of consecutive newly added zero slices does not change the value of max {IKI, ILl} and Bob can thus get slices of total size 10. In case 3, Alice first takes the slice of size a. The even number of added zero slices does not change the fact that before she is able to take any other nonzero slice, Bob takes either the slice of size a or the two slices of size c. After this, the value of max {IKI, ILl} is the same as in the proof of Claim 17. • For w = 0 or w = 1, the cutting used in Corollary 18 has slices of only three different sizes 0,1,2. If all the slices have the same size, then Alice always gets at least half of the pizza. But two different slice sizes are already enough to obtain a cutting with which Bob gets 5/9 of the pizza:

Claim 20. If Bob can make slices of only two different sizes, then he can gain 5/9 of the pizza by cutting the pizza into 21 slices of sizes 0 and 1 in the following way: 001010010101001010101. Moreover for any odd n 2: 21, there is a cutting with n slices of two different sizes of slices, for which Bob has a one-jump strategy with gain

5IPI/9. Proof. Let n 2: 21 and let the cutting be (00)mlOlOlOlO0l010010101 where m 2: 1. After Alice makes her first turn, there will be an even number of slices remaining and they can be partitioned into two classes based on their parity in the new sequence. If Alice takes a slice of size 0 in the first turn, necessarily one of the classes will contain more slices of size 1 than the other class. Bob's strategy will be to take the available element of the bigger class in each turn. Consequently Alice will be forced to take an element of the smaller class in each turn. If in the first turn Alice takes a slice of size 1 but the previous partitioning still gives two classes of different size, Bob uses the same strategy. Obviously Bob gains at least 5/9 of the pizza in these cases . There are three slices of size 1 indicated by bold which if taken in the first turn result in a partitioning into two classes of equal size: (OO)m1010101001010010101. The sequences below show the situation after the first turn, the vertical line indicates the place of the taken slice: 1. (00)m!1010IQ100101 P010101

Solution of Peter Winkler's Pizza Problem

81

2. (OO)m10101010q 1OIQ0101011 3. (OO)mlOlO1010~ 101OQI01011 Bob takes the underlined slice of size 0 in his first turn. He will make shifts until he gets to one of the borders of the boxes indicated above, then he makes a jump in his next turn and afterwards only shifts till the end of the game. Regardless of what turns Alice makes, depending on the case Bob will obtain at least one or two slices of size 1 before anybody reaches outside the given box. As he makes a jump in his next turn and later only shifts , he will get all slices of size 1 outside the box. Therefore in all cases his gain adds up to at least 5/9 of the pizza. The claim can also be checked for any fixed odd number n 2:: 21 with a computer program based on the algorithm from Section 7.2. •

4.

FIXED NUMBER OF SLICES

Here we prove Theorem 2. The theorem is trivial for n = 1 and easy for n even as observed in the introduction. Further, the theorem for n E {15, 17, 19, . . . } follows from Theorem 1 and Claim 17. An upper bound g(n) ~ 1/2 for any n 2 2 can be seen on the pizza 1100 .. . 00. It remains to show that Alice has a one-jump strategy with gain 1/2 for any pizza PIP2 . . . Pn, where n is odd and 3 ~ n ::; 13. Let n be odd and let 3 ::; n ::; 13. We partition the characteristic cycle V = VI V2 .. • V n into six arcs A, B , ... , E in the same way as in Section 2. We may suppose that each of the six arcs has a positive length, since otherwise Alice has a zero-jump strategy with gain 1P1/2 (by Claim 12). Therefore, as l(A) = l(D) + 1, 1( C) = l(F) + 1 and l(E) = l(B) + 1, and n ::; 13, at least one of the arcs B, D, F has length at most 1 (and hence, exactly 1). Due to the symmetries, it therefore suffices to prove the following claim:

Claim 21. If l(B) = 1 then Alice has a one-jump strategy with gain 1/2. Proof. First we describe a one-jump strategy with gain b+min {c+d, f+a} . Alice's first turn in the strategy is on the only slice of B . Recall that l(E) = l(B) + 1 = 2. Bob can choose between the two slices of E in the

82

J. Cibulka, J. KynCl, V. Meszaros, R. Stolar and P. Valtr

second turn. Alice takes the other slice of E in the third turn of the game . In the rest of the game, Alice makes only shifts, thus collecting slices of some pair of arcs Xl and X 2 such that X lEX2 is a half-circle (see Figure 6). Since none of the half-circles CDE and EFA can be replaced in the triple T by a half-circle of a strictly smaller size, the sum IXll + IX21 achieves its minimum either for Xl = CD and X2 = 0, or for Xl = 0 and X 2 = FA . Thus, the portion collected by Alice in the whole game is at least 91 :=b+min{c+d,f+a}.

By two applications of Observation 11, Alice also has a zero-jump strategy with gain 92 :=max{c+d+e,e+f+a}. One of the two strategies is a one-jump strategy with gain

5. CUTTINGS INTO 15 AND 21 SLICES Here we prove that Bob's cuttings described in Section 3 include all pizza cuttings into 15 and into at most 21 slices where he gets his best possible gain. Theorem 2 implies the minimality of 15 slices as well.

Claim 22. Corollary 18 describes, up to scaling, rotating and flipping the pizza upside-down, all the pizza cuttings into 15 slices for which Bob has a strategy with gain 5IPI/9. Proof. Suppose Bob cuts the pizza into 15 slices so that Alice cannot gain more than 4IPI/9. It follows that p(V) ~ 4/9 < 1/2. By Claim 12, the characteristic cycle can be partitioned into non-empty arcs A, B, C, D , E , F such that ABC, CDE and EFA form a minimal triple. Following the proof of Theorem 1, we assume without loss of generality that a+b+ c ~ c+d+ e ~ e+ f+a and we consider the same three strategies with gains 91, 92 and 93. Combining the assumption 91,92 , 93 ~ 41PI/9 with the inequalities (391 + 492 + 293)/9 ~ (41PI + a + c/2)/9 ~ 41P1/9 we get equalities everywhere. Consequently, 91 = 92 = 93 = 41PI/9 and a = c = O. This, in particular, implies that e + f = b/2 + e/4 + d = f /2 + b = 4IPI/9.

83

Solution of Peter Winkler 's Pizza Problem

Now we show that e = O. Applying Claim 21 three times, we get that = l(D) = l(F) = 2 and l(A) = l(C) = l(E) = 3. Particularly, the length of the circular sequence V' obtained by concatenating arcs Band E is 5. Hence, by Theorem 2, Alice has a strategy with gain b/2 + e/2 on V'. Following the proof of Claim 15 we get a two-jump strategy on V with gain b/2 + e/2 + c + d = tn + e/4 = 41PI/9 + e/4. Therefore e = O.

l(B)

It follows that f = 4IPI/9, b = 21PI/9 and d = 3IPI/9. If one of the two slices in B had size greater than 1P1/9, then by the proof of Claim 13 Alice would have a one-jump strategy with gain greater than 1P1/9 + c + d = 4IPI/9. It follows that both slices in B have size exactly IPI/9. Similarly, using an analogue of Claim 13, we conclude that both slices in F have size exactly 21PI/9 and both slices in D have size at most

2!PI/9. • Lemma 23. Let P = PIP2 ... Pn be a cutting of a pizza into an odd number of slices for which Bob has a strategy with gain 9 ~ 1P1/2. Let x = miniE{1,2,...,n} IPi I and let P' = p~p~ . .. p~ be a cutting of a pizza with slices of sizes Ip~1 = IPil - x. If x > 0, then for the cutting P' Bob has a strategy with gain strictly greater than gIPII/IPI. Proof. Let l: be Bob's strategy for the cutting P with gain 9 ~ 1P1/2. For the cutting P', Bob uses the same strategy l:. In this way he is guaranteed to get a subset Q' c P' of (n -1) /2 slices such that the corresponding subset Q c P has size at least g. Therefore, Bob's gain is

L p~EQ'

Ip~1 =

L

(Ipi\- x) ~ 9 - x(n -1)/2.

PiEQ

Since IPII = IPI - xn, we have to show that (g - x(n - 1)/2) IPI > g( IPI-xn) , which is equivalent to x(n-1)IPI/2 < xng. The last inequality follows directly from the assumptions 9 ~

1P1/2 and x > O.

•

Claim 24. Corollary 20 describes, up to scaling, rotating and flipping the pizza upside-down, all the pizza cuttings into at most 21 slices of at most two different sizes for which Bob has a strategy with gain 5IPI/9. Proof. Let P be a cutting of the pizza into n ~ 21 slices of at most two different sizes for which Bob has a strategy with gain 5IPI/9. If n is even, then Alice has a (zero-jump) strategy with gain IPI/2. If all slices have positive size, then by Theorem 1 and Lemma 23, Bob has no strategy with

84

J. Cibulka, J. KynCl, V. Meszaros , R. Stolar and P. Valtr

gain 5IPI/9. Therefore n is odd and at least one slice in P has size O. So we can without loss of generality assume that each slice has size a or 1. Now we proceed exactly as in the proof of Claim 22, up to the point where we are showing that e = O. After we apply Claim 21 (three times), we only conclude that l(B), l(D), l(F) ~ 2 and consequently l(A) , l(C), l(E) ~ 3. The length of the circular sequence V' obtained by concatenating arcs Band E is then at most 11. The rest of the argument that e = a is exactly the same as in the proof of Claim 22. We also conclude that f = 4IPI/9, d = 31PI/9 and b = 2IPI/9. Since the numbers f, d and b are non-negative integers , their sum must be a positive multiple of 9. Since l(F) + l(D) + l(B) :::; 12 and the size of each arc is bounded by its length, we have IFI = 9, f = 4, d = 3 and b = 2. Consequently, l(F) ~ 4, l(D) ~ 3, l(B) ~ 2 and l(C) ~ 5, l(A) ~ 4, l(E) ~ 3. Since l(A) + ... + l(F) :::; 21, none of these six inequalities may be strict. Therefore A = 0000, B = 11, C = 00000, D = 111, E = 000 and F = 1111. •

6.

ONE-JUMP STRATEGIES

The main aim of this section is to prove Theorem 3. The following corollary proves Theorem 3(a). Corollary 25. Alice has a zero-jump strategy for V with gain 1V1/3. The constant 1/3 is the best possible. Proof. The gain 1V1/3 trivially follows from Observation 11. Let V = 100100100. For every element v of V there is a half-circle C; of size not greater than 1V1/3 covering it. So no matter which element v Alice takes in the first turn, as Alice only makes shifts, Bob can play in such a way that Alice gets Cv . • In the rest of this section we prove Theorem 3 (b).

Solution of Peter Winkler 's Pizza Problem

85

6.1. Lower bound

In this subsection we show the strategy for Alice to gain at least 7/16 of the pizza. We can assume that the number of slices is odd and that p(V) < 1V1/2 (otherwise Alice has a strategy with gain 1P1/2 ~ 7IPI/16). We also fix a minimal triple of half-circles and a partition of V into arcs A, B , C, D , E and F given by Claim 12. We can use the zero-jump strategy with gain equal to p(V) and the one-jump strategies from Claim 13 (and its analogues). It can be shown, however, that these strategies alone guarantee Alice only 3/7 of the pizza. To improve Alice's gain we introduce one more one-jump strategy. Claim 26. Alice has a one-jump strategy with gain 3b/8+e/2+min {c + d, j + a} if p(V) < 1V1/2. Proof. If 3b/8 - e/2 :::; 0, Alice starts by taking a slice from E and then she makes shifts only. As observed in the proof of Claim 13, the potential of any slice in E (and thus Alice's gain) is equal to e + min {c + d, j + a} ~ 3b/8 + e/2 + min{c+ d,j + a}.

For the rest of the proof we assume that 3b/8 - e/2 > 0. The main idea of the Alice's strategy is to start with taking a slice somewhere in the arc B and to jump at some appropriate moment before crossing the boundary of B. Let k = I(B) . For i = 0,1 , .. . k, let B, be the initial sub arc of B of length i. Symmetrically, let B: be the arc containing the last i slices of B. Similarly we define arcs E; and E: for i = 0,1, ... k + 1. For i = 0,1 , ... k, let h(i) = IBil- IEil and h'(i) = IB:I-IE:I. The functions hand h' can be used to measure the difference between Alice's and Bob's gain during the first phase (before Alice decides to jump). We call this difference an advantage of Alice. During the first phase, Alice takes a sub-arc of B and Bob takes an equally long subarc of E. If Alice took B j \Bi and Bob took E j \Ei during the first phase, then Alice's advantage is h(j) - h( i). The other possibility is that Bob took Ej+1 \ E i+1 j equivalently, Alice took B:, \ Bj, and Bob took E i , \ Ej', where i ' = k - i and j' = k - j. In this case the advantage of Alice is h'(j') - h'(i'). By the minimality of the triple that determined the arcs A , B , ... , F , both functions hand h' are non-negative.

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J . Cibulka., J. KynCl, V . Meszaros, R. Stolar and P. Valtr

Similarly as in the previous strategies, Bob's best choice after Alice's jump is to let Alice take the rest of the arc E and one of the arcs CD or FA . It follows that if Alice's advantage is g, then her gain will be at least 9 + e + min {c + d, f + a}. It only remains to show that Alice can always achieve an advantage greater than or equal to 3b/8 - e/2 . Let i be the largest index such that h(i) :::; 3b/8 - e/2. Symmetrically, Let i' be the largest index such that h'(i') :::; 3b/8-e/2. We distinguish two cases.

Case 1: i + i' :::; k. Equivalently, B, and B~, are disjoint. Observe that we actually have i + i' :::; k - 1. Alice starts by taking any of the slices from B \ iB, UBi')' She jumps as soon as Bob takes the first or the last slice from E. During the first phase either Alice took B j and Bob took Ej for some j ~ i + 1, or Alice took Bj, and Bob took Ej, for some j' ~ i ' + 1. Alice's advantage is 9 = h(j) > 3b/8 - e/2 in the first case and 9 = h'(j') > 3b/8 - e/2 in the second case. Case 2: i + i' > k . Divide the arc B into consecutive arcs B I = Bk-i' , B 2 = B, \ B k-i, and B 3 = B~_i ' Similarly the arc E is divided into I 2 3 E = Ek-i'+1, E = e. \ Ek-i'+1 and E = ELi+I ' Since IE 2 1 ~ IE21-IB2 1 = -( h(i) + h'(i') - (b - e)) ~ b - e - 2(3b/8e/ 2) = b/4, we have min (IEII, IE31) :::; (e - b/4)/2. We can without loss of generality assume that jEll :::; IE31, hence jEll :::; (e - b/4)/2. Note that the arc B I (and hence E I) is non-empty, as IBII ~ IBII - jEll = b - e - (IB~,I-IE~,I) = b - e - h'(i') ~ 5b/8 - e/2 > b/4 > O. Alice now plays as in the proof of Claim 13, where B is replaced by B I and E is replaced by E I. That is, she starts with taking the median slice of B I and jumps as soon as Bob takes the first or the last slice of E I . In this way she gets an advantage 9 ~ IBII/2 _jEll = (IBII-IEII) /2 -IE II/2 = (b - e - h'(i')) /2II/2 IE ~ (5b/8 - e/ 2)/ 2 - (e - b/4)/4 = 3b/8 - e/ 2. •

Remark. By iterating the strategy from Claim 26 we obtain an infinite sequence 2;1,2;2 , ... of one-jump strategies, where 2;k+1 recursively uses 2;k in the same way as the strategy 2;1 from Claim 26 used the strategy 2;0 from Claim 13. These iterated strategies give better gain when the ratio bf e tends to 1. However, if the ratio b/e is smaller than 5/3, the strategies 2;i are beaten by the previous one-jump strategies that start outside the arcs Band E.

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Solution of Peter Winkler's Pizza Problem

Proof of the lower bound in Theorem 3(b). As in the proof of Theorem 1, we may without loss of generality assume that a + b + c ::; c + d + e ::; e+ f +a. By Observation 11, Alice has a zero-jump strategy with gain 91 := e + f

+ a.

By Claim 15, Alice has a one-jump strategy with gain 92 := b/2 + min {c + d, f + a} = b/2 + c + d.

By an analogue of Claim 15, Alice also has a one-jump strategy with gain 93 := f /2 + min {a + b, d + e} = f /2 + a + b. By Claim 26, Alice has a one-jump strategy with gain 94 := 3b/8 + e/2 + min{c+ d,J + a} = 3b/8 + e/2 + c+ d.

One of these four strategies gives gain max {91,92 ,93 ,94} 2:: (591 + 392 + 493 + 494)/16

= (9a + 7b + 7c +

7d + 7e + 71)/16 = (71 P I + 2a) /162:: 7IPI/16.

•

6.2. Upper bound Claim 27. If Alice is allowed to make only one jump, then Bob has a strategy with gain 9IPI/16. This gain is achieved for the following cutting of the pizza into 23 slices: 20200200202006060050500. Proof. The characteristic cycle V is depicted on Figure 9. The size of the pizza is 32, which means that we need to show that Bob can get slices with the sum of the sizes at least 18. The potential of the cutting is 14, thus Alice has no zero-jump strategy with gain greater than 14. It is easy to check that on the cutting sequence P , every zero slice has a neighboring nonzero slice and both neighbors of every nonzero slice are zero slices. If Alice starts by taking some zero slice, then Bob takes the nonzero

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J. Cibulka, J. KynCl, V. Meszaros, R. Stolar and P. Valtr

Fig . 9. Characteristic cycle of the cutting used in Claim 27

slice. Alice can then jump, but since she cannot make any more jumps, she would get only at most the potential of V. If she does not jump, she takes a zero slice in her next turn and from Observation 9, Bob can take one of the two arcs. Since Bob already took a nonzero slice, the sum of the sizes of the two arcs is at most 30. If both the arcs have size 15, then both must contain a slice of size 5 because there are no other slices of odd size. But this is impossible because the two slices with size 5 are neighbors on the characteristic cycle. Thus one of the arcs has size at most 14 and if Bob chooses the other one, he gets 18 for the whole game. Now we may assume that Alice starts by taking a nonzero slice. In his first turn, Bob takes a zero slice such that Alice cannot take a nonzero slice in the next turn. In the first phase , Bob makes shifts. The first phase ends after Alice's jump or if Bob's shift would allow Alice to take a nonzero slice in her next turn.

If Alice jumped in the first phase , she would get at most the slice from her first turn plus the potential of one of its two neighbors from the cutting sequence P. But it is easy to verify that this would mean a gain at most 14 for Alice. If Alice did not start in V4 or in VlO and did not jump, then it is easy to verify that after the first phase, one of the two Bob's zero-jump strategies from Observation 9 guarantees Bob gain 18. If Alice started in VlO and the first phase did not end by Alice's jump, then Bob continues making shifts until either Alice jumps or his shift would allow Alice to take the VIS slice in her next turn. This is the second phase. If Alice jumped during the second phase then Bob can make sure that she gets at most IVlOl + IVI41 + IV41 + IV51 = 12, see Figure 10 (left)) . If she did not jump until the end of the second phase , then from Observation 9,

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Solution of Peter Winkler's Pizza Problem

Bob can make sure that Alice gets at most IVIOI Figure 10 (right).

+ IVI41 + IV91

14, see

b)

a) Fig. 10. Two examples of games starting in

V10 illustrating how Bob can prevent Alice from gaining more than 14

If Alice started in V4 and the first phase did not end by Alice's jump, then Bob starts the second phase by making a jump and then only shifts. The second phase ends after Alice's jump or if Bob's shift would allow Alice to take V9. If Alice jumped during the second phase then Bob can make sure that she gets at most IV41 + Ivsl + IVISI + IVI91= 14, see Figure 11 (left). Otherwise we use Observation 9 to show that Bob can make sure that Alice gets at most IV41 + Ivs l + Ivai + IVI I = 8, see Figure 11 (right) . •

a)

b)

Fig. 11. Two examples of games starting in

V4 illustrating how Bob can prevent Alice from gaining more than 14

90 7.

J. Cibulka, J. Kyncl , V. Meszaros, R. Stolar and P. Valtr ALGORITHMS

7.1. Linear Algorithm In this section we describe an algorithm proving Theorem 6. For a given cutting of the pizza with n slices, the algorithm computes Alice's two-jump strategy with gain 4jPj/9 in time O(n). Without loss of generality we may assume that IVI is a part of the input. The algorithm first computes consecutively the sizes of all n halfcircles and finds a half-circle of minimum size in the following way. Consider the characteristic cycle V = VOVI ... Vn-l. For i = 0 to n - 1 let Si be the variable in which the size of the half-circle with starting point Vi, continuing in clockwise direction, is stored. Let S be the size of a currently minimal halfcircle H , and V the starting point of H. In the initialization step, compute So := l:;:~I)/2Ivjl and set S := So and V := Vo. Then for i = 1,2, .. . , n -1 , compute Si := Si-l -lvi-I I + hn+2i-I)/21. If Si < s , then set S := s; and The above computations can be done in time O(n).

V := Vi .

After t hese precomputations we get a half-circle H of minimum size that we fix. Let Vk and Vk+ (n-3 )/2 be the two uncovered elements neighboring H on V. In the following the algorithm computes the potentials of the elements of the uncovered arc X = Vk .. . Vk+ (n-3 )/2' Any half-circle covering an element Vi of X also covers Vk or Vk+(n-3)/2 ' Let the right potential of Vi be the minimum of the sizes of half-circles covering both Vi and v k+ (n - 3)/ 2' The algorithm computes the right potential Pr for Vk by comparing the values of Sk- l and Sk, i.e., Pr(Vk) := min {Sk-l, Sk}. For i = k + 1 to k + (n - 3)/2 set Pr(Vi) := min {Pr(Vi -I), Analogously let the left potential of Vi be the minimum of the sizes of half-circles covering both Vi and Vk. The computation of the left potentials PI is similar. Obviously the potential of Vi is min {PI(Vi),Pr(Vi)}' The computations are done in time O(n) .

sd.

The potential of any element of X is at least as big as the potential of any element of H due to the minimality of H . Therefore p(V) is equal to the maximal potential on X. If p(V) ~ IVI/2, then the algorithm returns an element of potential p(V) in time O(n). This will be Alice's first turn and all her other turns will be shifts that can be computed in time 0(1). From now on we assume that p(V) < IVI/2. The algorithm finds the index j E X for which PI(Vj) + Pr(Vj) is minimal among all j such that

Solution of Peter Winkler's Pizza Problem

91

both PI (Vj) and Pr(Vj) do not exceed p(V). There exists such a j E X as a consequence of Claim 10 and Claim 12. Let HI be the half-circle that gives the left potential PI(Vj) for Vj, and H 2 the half-circle that gives the right potential Pr(Vj) for Vj' Then H , HI and H 2 form a minimal triple. Indeed, suppose that there is a half-circle in the triple that can be replaced by a half-circle of strictly smaller size. Clearly, this half-circle is not H but HI or H2. A contradiction to the choice of j . We get that the triple is minimal. Knowing the minimal triple the algorithm computes A, B , G, D, E , F and V' in time O(n). If p(V') 2': lV'I/2, a slice of potential p(V') can be found in time O(n) . Otherwise, the algorithm computes the arcs A', B' , G', D', E', F' on V' similarly as above in time O(n) . A median slice of B' can be found in time O( n) by traversing B' twice. At first the algorithm computes IB'I. Then it adds the sizes of the elements one by one again and checks in every step if the sum exceeds IB'I/2. That will occur at a median slice. The algorithm orders a+b+c, c+d+e and e+ f +a. Assume without loss of generality that a + b + c ::; c + d + e ::; e + f + a. According to Section 2.5 Alice has three strategies with gains gi := e + f + a, g2 := b/2 + e/4 + c + d and g3 := f /2+cJ.4+a+b . These strategies were computed in time O(n) as described above. Alice chooses one of the three strategies corresponding to max {gI ,g2,g3}. Once the strategy is known, Alice's turn can be computed in time 0(1) in any position of the game.

7.2. Optimal strategies The following result implies, for example, that Alice can be forced to make only jumps (except AI, An) in her optimal strategy. Observation 28. (i) For any odd n 2': 3 and for any of the 2n - 2n permutations allowed in the game on n slices, the pizza can be cut into n slices in such a way that if both Alice and Bob make only optimal turns then the order of taken slices is the chosen permutation.

(ii) For any even n 2': 4 and for any of the 2n - 3n plausible permutations in which Alice's second turn is a jump, the pizza can be cut into n slices in such a way that if both Alice and Bob make only optimal turns then the order of taken slices is the chosen permutation. Proof. We give the sizes 1,1/2,1/4, . .. in the order in which we want the slic es to be taken.

_

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J. Cibulka., J. Kyncl, V . Meszaros, R. Stolar and P. Valtr

Now we describe an algorithm that computes both player's optimal strategy for a given cutting of the pizza with n slices in time O(n 2 ) . Proof of Claim 7. A position of a game is an arc X characterized by its leftmost slice x; and its rightmost slice x; or the empty-set if there are no more slices left. If l(X) = 1, then x; = X r . There are n 2 - n + 2 possible positions X. The parity of l(X) determines whose turn it will be. For i = 0 to n - 1 the algorithm traverses all X with l(X) = i and decides the best strategy for the player on turn. The best possible gain on X is the value of X , denoted by v(X). The algorithm stores v(X) for all positions X . For i E {O, I} the strategy is obvious and v(X) = IXI. Let X - X be the arc X omitting the slice x. For i ~ 2, v(X) = IXI- min {v(X - Xj),v(X - xr ) } and the player takes the corresponding slice yielding the minimum in the previous expression . All this can be done in time O(n 2 ) . A program implementing the algorithm described above can be downloaded from the following webpage: http://kam.mff.cunLczrcibulka/pizza. The program can be used to verify Claims 17, 20 and 27. •

Acknowledgments. We thank Peter Winkler for posing this problem and for the discussions. The third author thanks Dirk Oliver Theis for the discussions about the upper bound during Fete of Combinatorics and Computer Science in Keszthely, August 11-16, 2008. We also acknowledge the use of the LPSolve IDE [2J software , which made our computations much more efficient.

REFERENCES

[IJ J. Cibulka , J . Kyncl, V. Meszaros, R. Stolar and P. Valtr, Generalizations of Peter Winkler 's Pizza Problem, in preparation. [2] H . Gourvest , LPSolve IDE v5 .5, http://www.progdigy.com

M. Berkelaar, J. Dirks, K. Eikland and P. Notebaert, lp...solve 5.5, http://lpsolve.sourceforge.net/5.5 [3]

K. Knauer, P. Micek and T . Ueckerdt, How to eat 4/9 of a pizza, manuscript.

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Solution of Peter Winkler's Pizza Problem

Josef Cibulka, Rudolf Stolar Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostronske tuim . 25, 11800 Praha 1, Czech Republic e-mail: cibulka~kam.mff.cuni.cz ruda~kam.mff . cuni . cz

Jan Kyncl, Viola Meszaros, Pavel Valtr Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University , Faculty of Mathematics and Physics, M olostromske tuim . 25, 11800 Praha 1, Czech Republic e-mail:

Viola Meszaros Bolyai Institute, University of Szeged, Aradi vertanuk tere 1, 6720 Szeged, Hungary e-mail:

viola~ath.u-szeged.hu

kyncl~kam.mff.cuni .cz

Fete of Combinatorics, pp. 95-137.

BOlVAI SOCIETY MATHEMATICAL STUDIES, 20

TIGHT BOUNDS FOR EMBEDDING BOUNDED DEGREE TREES

BELA OSABA*, IAN LEVITT, JUDIT NAGY-GYORGY and ENDRE SZEMEREDIt

Let T be a tree on n vertices with constant maximum degree K . Let G be a graph on n vertices having minimum degree o(G) ~ n/2 + CK logn, where CK is a constant. If n is sufficiently large then T C G. We also show that the bound on the minimum degree of G is tight.

1.

INTRODUCTION

In this paper we will only deal with simple graphs, without loops or multiple edges. Given a graph F = (V, E), and U C V, the subgraph of F induced by the vertices of U will be denoted by Flu' The degree of a vertex v E V(F) is denoted by degF(v), or by deg (v) if F is clear from the context. The minimum degree and maximum degree of F will be denoted by 8(F) and 6.(F), respectively. We will let N(v) represent the neighborhood of a vertex v, hence deg (v) = N (v) The number of vertices in a graph F is written v(F) , and the number of its edges is e(F) . If A, B c V(F) , then e(A, B) is the number of edges of F with one endpoint in A and the other endpoint in B, and e(A) = e(A , A) = e( FIA) ' For a tree T we denote the set of its leaves by £(T).

I

I.

Given a bipartite graph H , a proper 2-coloring of H is a 2-coloration of the vertices of H such that adjacent vertices of H get different colors. "Partially supported by th e Office of Sponsored Programs at WKU and the OTKA grants T049398 and K76099. tpartially sup p orted by the Ellentuck Fund.

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B. Csaba, I. Levitt, J . Nagy-Gyorgy and E. Szemetedi

For graphs J and F , we will write J c F if and only if there exists an adjacency preserving injection I : V{J) -+ V{F). If I{x) = v then we say that x maps onto v and that v is covered by x, If a vertex of F is not covered by any vertex of J then we call it uncovered. Throughout the paper we use a « b to denote that a is sufficiently smaller than b. Our goal is to show the following:

Theorem 1. Assume that T = (V, E{T)) is a bounded degree tree on n vertices with f1{T) = K . Let G = (W,E{G)) be a graph on n vertices. Assume that its minimum degree , <5{G) ~ n/2 + CK logn where CK is a constant depending only on K . Then there exists no such that T C G for n ~ no . Furthermore, the bound on <5 (G) is tight: there exists a graph G with <5{ G) > n/2+ log n/17 such that the complete ternary tree on n vertices is not a subgraph of G if n is sufHciently large. Bollobas [2] conjectured that if G is a graph on n vertices, <5{G) ~

{1/2+c)n for some e > 0, and T is a bounded degree tree on n vertices, then T C G. The problem was solved in the affirmative by Komlos , Sarkozy and Szemeredi [3] for large graphs. They then strengthened their result (see [4]), and showed, that f1{T) need not be bounded: there exists a constant c such that T C G if f1{T) ::; cn/logn, <5{G) ~ (1/2 + c)n and n is large . Both proofs are based on the Regularity Lemma - Blow-up Lemma Method. In this paper we eliminate the use of the Regularity Lemma under the original assumption that f1{T) is bounded. As a result the threshold no is much smaller, but more importantly, we prove a stronger statement than the original conjecture. Indeed, in Theorem 1 <5{G) - n/2 can be as small as O{logn), and o{logn) is not sufficient. We may assume that G is minimal with respect to edge deletion, retaining the property that <5{G) ~ n/2 + CK logn. In particular, there are no edges xy E E{G) such that both vertices x, y have degree larger than n/2 + CK logn. We divide the problem into two subproblems, depending on whether G is extremal or non-extremal. We call G ,-extremal for, > 0 if either G or its complement G contains a subgraph on n/2 vertices with at most ,n2 induced edges. Otherwise G is -y-non-cxtremal. Let, = K- 20 . We first prove the statement for ,-non-extremal graphs, then for -y-extremal graphs.

In each case we will construct the adjacency preserving mapping I : V -+ W step by step. It is worth noting that while at any time in the course of embedding we have an adjacency preserving partial mapping, the

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Tight Bounds for Embedding Bounded Degree Trees

final I is not necessarily an extension of it . It is possible that we rethink the way we mapped certain subtrees. In the last section we show that for complete ternary trees the log n additive term in the minimim degree of G is necessary in case G is a suitably defined extremal graph. Let us remark that studying some cases of trees having larger degrees suggest that one needs the minimum degree <5(G) 2: n/2 + C.6.(T) logn, up to the point when the tree has maximum degree en/ log n for some positive constant c. Throughout the paper we make no attempt to optimize on the absolute constants.

2.

THE NON-ExTREMAL CASE

The case of non-extremal G is divided into two sub cases depending on the structure of T. It is shown below that T has either a small subtree with many leaves, which we call a broad subtree , or it has a large subtree with few leaves, called a long subtree. Specifically, Tb is a broad subtree of T if £(Tb) c £(T), V(Tb) ~ ,5 n /K2 and 1£(n)1 2: ,7n. We call 11 c T long if £(11) c £(T), v(1I) > ,5 n /K3 and 1£(11)1 < ,7 n . We use a simple method to decide which case applies.

2.1. Some tools for the proofs in the non-extremal cases

Definition 1. Assume that we are given a rooted tree F with root p. Let x and y be any two vertices of F. We say that y is below x if the simple path connecting y with p goes through x . Let F(x) denote the subtree rooted at x containing every vertex which is below x. Sometimes we will call x the tip of the subtree F(x). We turn to the method of determining whether T has a broad or a long subtree. First, (temporarily) designate a root p E V(T) . Repeat the following for every x E V(T): color x blue if 1£(T(x)) 2: n , otherwise color it red . If p is red, then T itself is long. If not, then T must have a blue vertex x such that all of its children are red. If v(T(x)) ~ ,5 n/ K 2 then T(x) is a broad subtree and we let Tb = T(x). If v(T(x)) > ,5n/ K 2

I

,7

98

B. Csaba, 1. Levitt, J. Nagy-Gyorgy and E. Szemeredi

then one of its red children, y, is such that long subtree, and we let 1/ = T(y) .

,5

v(T(y)) > n/ K3. T(y)

is a

,n

Observe that if G is not ,-extremal, then it has many triangles: there 2 edges in the neighborhood of every vertex, since these are at least neighborhoods are of size larger than n/2. As any such edge is in a triangle, summing over vertices will count each triangle three times. We find that 3 there are at least /3 triangles in G.

,n

We record here a simple statement, which is also very useful throughout the proof:

Lemma 2. Let H be a tree of size m. If J is a graph with fJ(J) ~ m, then there is an embedding of H into J . If J' = J'(A, B) is a bipartite graph such that every bE B has at least m/2 neighbors in A and every a E A has at least m neighbors in B then H C J' . Proof. We leave the proof of the first part to the reader. For the second part we remark that H is 2-colorable, and one of its color classes has size at most m/2. _ We will frequently apply the following folklore statement (proof is omitted):

Lemma 3. Every graph H has a subgraph H' such that

fJ(H')

~

e(H)/v(H).

_

Remark 1. We will frequently use the above two lemmas in the following way: whenever we find a dense subgraph in the uncovered part of W, we can find a subgraph of it with large minimum degree into which we can map a large subtree of T . Let F ments:

= F(A, B) be a bipartite graph satisfying the following require-

• IAI = t •

')'-3

and

IBI = t 2 ,

«t,

• every bE B has at least (1/2 + ')'3)t neighbors in A. Then we have the following Cleaning Lemma.

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Tight Bounds for Embedding Bounded Degree Trees

Lemma 4 (Cleaning lemma - first version). F has a subgraph F' = F'(A', B') such that A' c A, B' c B, every s « B' has at least (1/2+,3/2)t neighbors in A' and every a E A' has at least t neighbors in B' . Proof. First we assume, that every vertex in B has exactly (1/2 + ,3)t neighbors in A - if necessary, we discard edges incident to those vertices of B which have larger degrees. Then we have e(F) = (1/2 + ,3)t3. We will find the desired subgraph F' step-by-step in the following way. Let Al = {a E A : IN(a)nBI < t}, and let B 1 = {b E B : N(b) n (A - AI) (1/2 +,3 /2)t}. In the first cleaning step we delete the vertices of Al from A and the vertices of B 1 from B. Removing these vertices deletes at most IA 11t + 2IA 1It(1/2 + ,3);'3 edges from F. In the ith cleaning step we first identify a subset Ai C A - u~-=-=i A j and

I

I ::;

a subset B, C B - u~-=-=i B j :

and

e(A-t , B - Ui.- 1 B ·) .I < 2 IB t

-

,

3t

)=1)

<

t 2lA-I _

,

3'

Deleting the vertices of Ai and B, therefore removes at most IAiit + 2IAil(1/2 + ,3)t/,3 edges. Let us assume that after k cleaning steps the cleaning process stops: either every vertex left satisfies the degree requirements of the lemma, or there are no vertices left. The total number of edges we lose from F is at most

We must have the former case, as most of the edges are still present when the process stops. The induced subgraph on A-U~ Ai and B-U~ B, will be denoted by F', and is easily seen to satisfy the requirements of the lemma.

•

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B. Csaba, 1. Levitt, J. Nagy-Gyorgy and E. Szemetedi

For the second version of the Cleaning Lemma we assume that F satisfies the following: • IAI = t and IBI = t 3 ,

•,-4 «t, • every b E B has at least (1/2 - ,4)t neighbors in A . While the setup is somewhat different, the proof of the Lemma is very similar to that of the first version, we omit the details: Lemma 5 (Cleaning lemma - second version). F has a subgraph F' = F'(A', B') such that A' c A , B' c B, every bE B' has at least (1/2 - 2,4)t neighbors in A' and every a E A' has at least t 2 neighbors in B'. •

Randomly choose three disjoint subsets from W , denoted by D 1 , D 2 and M, with sizes ID11 = ID21 = ,8n, and IMI = ,IOn. The common size of the DiS will be denoted by 8. We make three assertions regarding these sets: • Every u E W has at least 8/2 - O( Jnlogn) neighbors in the and IMI/2 - O( In log n) neighbors in M with high probability.

DiS

• If G is ,-non-extremal, then the subgraphs induced by D 1 , D2 and M

will be , /2-non-extremal with high probability. • The induced bipartite subgraph on Dl and D2 will be , /2-nonextremal with high probability. That is, if A c D 1 , B C D 2 , each of size 8/2, then there are at least ,8 2/2 edges between A and B. The proofs of these statements are straightforward. We sketch a proof of the non-extremality of Ds, We demonstrate that the induced subgraph on D1 cannot have a sparse subset of size 8/2. The analogous statement, that the complement of the above induced subgraph cannot have a large subset with a few edges can be proved similarly. It is a simple exercise that if G is not extremal, then any set U for which Va,b E U, IN(a)UN(b)1 ~ (1/2+,)n cannot contain more than (1/2-,)n vertices. Conversely, if G were extremal, it would contain a set U of size n/2 and an element a E U such that IN(a) U N(b)1 :::; (1/2 + ,)n for all but at vertices b E U. Let A C D; be any set of size 8/2. If:Ja E A such most that IN(a) U N(b) I :::; (1/2 + ,)n for most b E A then with high probability about half the vertices of W (G) would have almost the same neighborhood

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Tight Bounds for Embedding Bounded Degree Trees

101

as a has. But that would imply the extremality of G. Thus, with high probability, a set A C D 1 of size 8/2 will have the property that for many pairs of vertices a, b E A,

I(N(a)UN(b))

nD11

~ (1/2+,)8-0(Vn log n ) .

It follows that for most pairs of vertices, I(N(a)UN(b)) nAI ~ ,8/2. Thus A is dense. Lemma 6. Let G be a non-extremal graph, and D 1 , D 2 be as above. Then there is a perfect matching in the bipartite subgraph of G induced by D 1 and D 2 •

Proof. We will check the Konig-Hall conditions. Since every u E W has at least 8/2-0(n) neighbors in D2, every He D 1 such that IHI :::; 8/2-,28 will have a neighborhood of size at least IHI in D 2 . Assume that IHI > 8/2-,28. If D2 had a subset H of size 8/2 - ,28 such that e(H, H) = 0, then the induced bipartite subgraph on D 1 and D2 would be extremal - if we complete Hand H by adding at most ,28 new vertices to both, the number of newly added edges is not more than 2,282 « ,8 2/2. Hence every H C D 1 having size at least 8/ 2 - ,28 has a neighborhood of size at least 8/2 + ,28 in D 2. We finish by noticing that every vertex in D2 has 8/2-0(n) neighbors, therefore, if H c D 1 is larger than 8/2 + o(n) then it is neighboring with the whole D 2 . • The following corollary is used to handle a small modification of the bipartite subgraph from Lemma 6.

Corollary 7. Discard at most ,2 8 vertices from D2, replacing them with arbitrary vertices which all have at least IDd/2 - o(n) neighbors in D 1 . Then the resulting new graph has a perfect matching. Proof. The proof of Lemma 6 is based on two facts: the first is that every vertex in the bipartite graph has at least 8/2 - o(8) neighbors in the other color class, the second is that the bipartite subgraph induced by D 1 and D 2 is non-extremal. Since every vertex of G has at least 8/2 - 0(8) neighbors in D 1 , if we replace ,2 8 vertices of D 2 by any ,2 8 vertices, the first condition will remain satisfied. For the second observe, that the modified bipartite graph will be , /3-non-extremal, since we lost at most ,2 8 2 edges between any two subsets of size 8/2 when discarding ,28 vertices from D 2 • •

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Lemma 8. Assume that G is non-extremal and u, v E W . Then there are at least ,n/5 vertex disjoint paths of length 3 connecting u and v.

I

Proof. Set a = N(u) n N(v)l . If a ~ n/2 - ,n/2 then, by the nonextremality of G, there are at least ,n2/2 edges in N(u) n N(v) . One can therefore find ,n/2 vertex disjoint edges in N(u) n N(v), determining ,n/2 vertex disjoint paths of length 3 between u and v. At the other extreme, if a ~ ,n/2 then there are at least ,n2 edges in N(u) induced by G, again since G is non-extremal. If v'is the endpoint of an edge in GIN(u)' then it has a neighbor in N(v). It follows that there are at least ,n2 /2 edges between N (u) and N (v), and we can easily find the vertex disjoint paths of length 3 between u and v. Assume now that ,n/2 ~ a ~ n/2 - ,n/2. Then the number of edges connecting N(u) n N(v) with N(u) U N(v) is at least a(n/2 - a) ~ ,n2/5. From these one can easily choose ,n/5 vertex disjoint edges to yield the desired vertex disjoint paths of length 3. •

2.2 . The first non-extremal case: T has a broad subtree 2.2.1. Preparations Finding distinguished leaves. Recall that neT is a broad subtree with £(Tb) c £(T), I£(n) I ~ '-y7 nand v(n) ~ n/ K 2. We choose the tip of Tb to be the root ofT, and denote it by r , As T has bounded degree, we can easily find s = n subtrees of Tb' denoted by {FI' F2, ... , Fs}, such that each F; has depth 4, and for each i, i . the dist ance between Fi and Fj is at least 4. Furthermore, for each 1 ~ i ~ s we distinguish a leaf ai E £(Fdn£(Tb) which is at depth 4. The immediate ancestor of ai is denoted by P(ai) and the root of Fi is called A(ai)' Observe that P(ai) is distance three from A(ai)' Denote the set of distinguished leaves by S = {aI, a2,' .. , as}. We will map P(al), P(a2),"" P(a s), aI, a2, ... ,as in a special way. In particular, we will map the P(ai)s to the vertices of D I , and most of the aiS to D2.

,5

,8

Embedding Ti: We start the embedding of T by that of Ti: We first map the A(ai)s into W - (D I U D2 U M) arbitrarily, and the P(ai)s into D I arbitrarily. Using Lemma 8 we find vertex disjoint paths of length 3 connecting I (A(ai)) with I ( P(ad), such that these paths avoid D2 and M , and contain vertices from DI only as an endpoint. We then finish the embedding of Ti; avoiding M U D2. Since Tb is very small and the minimum

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Tight Bounds Eor Embedding Bounded Degree 'frees

degree is large in G\ (M U D2) , we can easily embed in a bottom-up fashion from the A(ai)s towards r and top-down from the A(ai)s towards the leaves. We delay the embedding of S until the end. Observe, that with the above we have determined I on V(n) - s.

n.

n

2.2.2. Decomposition of T Our goal is to decompose T into subtrees of constant size such that most of the vertices are in good subtrees. We call a subtree in the decomposition good if there are at most two edges which connect it to other subtrees. We make the following claims, whose proofs are left to the reader: Claim 9. Let F be a tree. The number of vertices of F having degree at most two is at least v(F)/2. Claim 10. Let F be a tree. There are at most :(~J vertices which have degree at least a . The basic building block of the decomposition algorithm is simple. Observe that given a rooted tree F with maximum degree K and an integer h ~ v(F) one can find a vertex x E F such that h ~ v(F(x)) < Kh. Decompose F by trimming off subtrees recursively as long as there are at least (K + 1)h vertices not yet part of the decomposition. When the process stops, the remaining connected component comprises the final subtree in the decomposition. We call such a decomposition of F into subtrees all having size in the range [h, (K + l)h] an h-decomposition. Construct an auxiliary tree F whose vertex set is the set of the subtrees in the decomposition of F, and for which two vertices are connected iff there ~ an edge between the corresponding subtrees. By applying Claim 9 to F, we conclude that at least v(F)/(2(K + 1)) vertices of F are in good subtrees. We want to discard those subtrees of the decomposition which have many neighbors in F. Call a subtree bad if the corresponding vertex has degree larger than 2K,-10 in F . By Claim 10 F has at most v(F)/(2K,-10-2) bad subtrees. It follows that the total number of vertices of F in bad subtrees is at most v(F )/ 2.

,lO

Set m = 3(K + 1) log,-lO and let hI > h2 > ... > hm be a sequence of integer constants, such that hI = ,-40(4K3, - IO) m, and hi/hi+! = 4K 3 , - 1O . For the full decomposition of T - Tb' we iteratively determine hi-decompositions of smaller and smaller subtrees. In the process, we define a sequence of connecting edge sets, the first of which is E I = 0. First, find

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n

an hI-decomposition of T and take the union of T - Tb with E 1 . The set of subtrees in the hI-decomposition is divided into three sets - the good, the bad, and the active. Good and bad subtrees are as defined above, and the remaining subtrees are called active. Let Ez be the edges connecting the subtrees of the decomposition. For each active subtree F we find an hz-decomposition of F . We determine the hz-good subtrees of this decomposition as those which are connected to at most 2 other subtrees of F, noting that they may also be connected through edges of Ez to the larger decomposition . The bad subtrees determined by the decomposition of F are those which are connected to at least 2K ,-10 other subtrees, either within F or through E z. The rest of the subtrees are considered active for the next iteration. Having found an hz-decomposition of all of the active subtrees, define E3 to be the edges connecting all of the subtrees thus far determined - i.e. E3 is the union of Ez and the edges connecting the subtrees of the all of the hz-decompositions of the active subtrees from the previous step. . In general, after finding the hi-decomposition of every active subtree from the hi_I-decompositions and distinguishing the hi-good, the bad, and the active subtrees as before, we define E; to be the union of E i- 1 and the set of edges connecting the subtrees of the hi-decompositions. We stop after the m th iteration. We are interested in subtrees that are good in the entire decomposition . We claim that the number of vertices in such good subtrees is at least (1 - ,1O)v(T - Tb) ' Indeed in the i t h step, having decomposed an active subtree into N subtrees, at least N /2 of them are hi-good. Since E; has at most 2K,-10 edges which are adjacent to the decomposed active subtree, at most 2K,-lO of the hi-good subtrees out of N/2 are not good overall. By the definition of the hi at least N /3 subtrees will remain good, and these will contain at least hd3(K + 1) vertices of the decomposed active subtree. That is, at every step we gain a proportion of 1/ (3(K + 1)) for the new vertices in good subtrees. Since (1-1/(3(K + 1)))m::; ,10/2, in the ~nal decomposition at most ,1Ov(T - Tb) vertices are not in good subtrees, and every subtree has size between ,-40 and ,-40(4K3,-1O) "'. Certain good subtrees won't be easy to handle later. Consider a good subtree T which is connected to two other subtrees in the decomposition such that these two connecting edges have a common endpoint x in T. In order to eliminate this problem we do the following. First, delete T

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Tight Bounds for Embedding Bounded Degree Trees

from the set of good subtrees. Let Yl, Y2, . .. ,Ys denote the neighbors of x in l' and let T(Yl) ,T(Y2),. , . ,T(ys) denote the subtrees of l' originating from Yl,Y2,."'Ys, respectively. Ifv(T(Yd) > ,-20 then T(Yi) will be a good subtree, otherwise it will be bad . Since v(T) 2 ,-40 and at most (K - 1),-20 vertices are put into the set of bad subtrees, we still have at least (1 - ,IO)v(T - n) vertices in good subtrees.

n

by 1'1,1'2, ... ,Tl. Each good subtree Denote the good subtrees of T i; has bounded size, there are at most two edges which connect the good subtrees to any other subtree in the decomposition, the endpoints of these connecting edges always differ, and the vast majority of the vertices are in the union of the good subtrees.

2.2.3. Sketch of the embedding algorithm. We map large subtrees of T - Tb one-by-one using the Main Mapping Procedure, discussed below. This procedure works smoothly until about half of T is mapped at which point we are faced with problems satisfying the key constraint of the Main Mapping Procedure, namely that the uncovered vertices of G contain a large subgraph with a non-negligible minimum degree. If the set of uncovered vertices is too sparse, then we rethink the way we mapped certain subtrees. We will rearrange the mapping of some subtrees in such a way that we use many uncovered vertices, and the new set of uncovered vertices contains a large dense bipartite graph, which we use to continue the embedding of T.

n

When at most ,IOn vertices are left out from T - Ti, we can easily finish the embedding.

2.2.4. Finishing the embedding of T. Let us assume that we have already mapped most of T, only the vertices of S and a few subtrees, Tl' T 2 " , . , Tk are left unmapped, such that I U Tjl ~ ,2 8. Set T* = T (UTjUS). We embed the Tj into D 2 , easily accomplished as D 2 is a randomly chosen subset of W, and ID2 1 = 8 ~ ,2 8. A perfect matching between D; and D2 - I(UTj) U (W - (I(T*) U D2)) is ensured by Corollary 7 and determines the rest of the embedding. Indeed , as D; is covered by the P(ad, we map ai to the vertex matched to I(P(ai)). Hence, if we can map most of T such that we avoid D 2 , we can finish the embedding of T. In the leftover of this section we will work so as to achieve this goal.

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2.2.5. The Main Mapping Procedure. The method below is used to map large chunks of T into G one-by-one. We apply it throughout the embedding. Assume that we are in t~e course of the embedding. Suppose that the leftover of T has a subtree T of the following type: its root is the already mapped vertex y, y's children are Yl, Y2, . .. ,Ym (m ::; K), the children of the YiS is the set {Tl' T2, " ., Td, and there is a tree T j of constant size originating at Tj for every 1 ::; j ::; f . Denote by H a subset of those vertices of W - (M U D1 U D2) which have not yet been covered by a vertex of T. Assume that every vertex of H has at least v ('T) neighbors in H . Then we claim the following:

,-2

Lemma 11. Assume that the above conditions are satisfied. If M has at least (1-,/10)IMI vacant vertices, then I can be extended to the vertices of T such that I(Yi) E M, I(Tj - Tj) C H, and I(Tj) EM for every 1 ::; i ::; m and 1 ::; j ::; f . Proof. Let us denote the set of already covered vertices of M by X, then IXI ::; ,IMI/lO. Since M is a randomly chosen set, IN(I(y)) n(M -X)I ;::: (1/2 - ,/9) IMI. Let us define the set U = {u : u EM-X, IN(u) nHI

;::: ,2IHI}.

Again, as M is randomly chosen, every vertex of H has at least (1/2 ,/9)IMI neighbors in M - X. Simple computation shows that lUI ;::: (1/2 - ,/8)IMI . Since the induced subgraph on M is ,/2-non-extremal, there are at least ,IMI 2/5 edges in between U and N(I(y)) n (M - X) . Then we map the YiS to such vertices in N(I(y)) n (M - X) which have at least K 2 neighbors in U. If YiTk is an edge in T, then Tk will be chosen among the neighbors of I (yd . It is possible to embed the remaining small Tj subtrees in H due to the assumed minimum degree of H. •

Remark 2. We remark that when embedding such a subtree, we cover less than K 2 vertices from M . Therefore, if one embeds at least ,-12 vertices in each application of the Main Mapping Procedure, less than ,IMI/lO vertices of M will be covered throughout the embedding. Remark 3. We will apply the Main Mapping Procedure throughout the embedding. Whenever the set of vacant vertices contains a large dense subgraph, then by applying Lemmas 2 and 3 we can embed a new subtree from the leftover of T.

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Tight Bounds for Embedding Bounded Degree Trees

2.2.6. The Second Mapping Procedure. Let Q denote the set of those vertices of W - (M U D1 U D2) which have not yet been covered by a vertex of T . In the Main Mapping Procedure we needed that every vertex in H has many neighbors in H. This time we allow Q to be very sparse, but assume that IQI > ,.,l°n. The goal is to replace some of the covered vertices of G by those in Q in such a way that we gain a dense subgraph in the new Q. Let T' c T denote the portion of T that has already been embedded. Assume that T' has a subtree T with the following properties: (i)

T as a subtree of T'

(ii)

ITI = t, where ,-20:::; t :::; hI = ,-40( 4K3,-10) 30(K +l ) log

has exactly one neighbor in T' -

T; ')'-1 ;

(iii) The number of vertices of Q with degree at least (1/2 + ,3)t into at least t 2 .

T is

If the above is satisfied, we call T a remappable subtree. Notice that if there are many edges from Q to a good subtree T such that T is connected to T' by one edge, then T is remappable.

I

Let Z = {z E Q : N( z) n I(T)I ~ (1/2 + ,3)t}, and construct the bipartite graph F = F(A , B), where A = I(T) and B c Z such that IBI = t 2 . We connect a E A and b E B by an edge if they are adjacent in G. We apply the first version of the Cleaning Lemma to find a subgraph F'(A', B') c F(A, B) such that A' c A , B' c B , every b E B' has at least (1 + ,3)t/2 neighbors in A' and every a E A' has at least t neighbors in B'. We apply Lemma 2 to find a mapping of Tonto F' such that at most t/2 vertices of the new mapping cover vertices from A = I(T) and the rest cover vertices from B. We re-embed T using vertices from M as in the Main Mapping Procedure, but we do not map a new subtree seemingly it is a loss. On the other hand, there is a leftover bipartite graph F"(A" , B") c F(A, B) , where A" c A and B" C B, IA"I ~ t/2 and IB"I ~ t 2 - t, and every vertex of B" has at least ,3t neighbors in A". This dense bipartite subgraph can then be used to map a new unmapped subgraph of T by applying the Main Mapping Procedure. As is discussed in Remark 3, by Lemma 3 we are able to use F" in order to map a subtree of size at least ,-17/2. That is, we cover overall at most 2K2 vertices of M by the two applications of the Main Mapping Procedure, but map a new subtree which is of size at least ,-17/2. The restrictions on the speed with which vertices of M are mapped are respected.

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Good subtrees in the decomposition of T - n may be connected to two other subtrees. Therefore we consider substituting (i) by (i')

T c T'

has exactly two neighbors in T' -

T.

We show how to remap such a subtree. First, denote the two vertices of T which have neighbors in other subtrees by x and y, recalling that x =J y . Denote the neighbors of x and y in the adjacent subtrees by Zl and Z2 , respectively. If the distance between x and y is at most three then we map the whole path connecting x and y onto vertices of M such that I(x) is adjacent to I(zt} and I(y) is adjacent to I(z2)' This is easily done since M is a randomly chosen set. Then we map the subtrees of T below the vertices of the path by the Main Mapping procedure, using at most 6K 2 vertices from M. Otherwise denote the vertices of the path P connecting Zl and Z2 by Zl, x, ... , y", y', y, Z2. We map x and the part of T which is below P by the Main Mapping procedure, except the vertices y' and y and the subtrees which are below them. We map y" to any vertex in Q. Since M is nonextremal, it is easy to find a path of length three connecting y" and Z2 such the images of y' and y belong to M. Then we map the subtrees below y' and y by the Main Mapping procedure. Again, we use at most 6K 2 vertices from M and as above we do not cover the vertices of M too quickly. 2.2.7. The Third Mapping Procedure. As in the second mapping method we assume that Q is very sparse and IQI > ,IOn. Recall that the good subtrees of the decomposition are denoted by Ii, T2' ... ,Te. Let ti = v(ID. Assume that only a few vertices in Q have many neighbors in the good subtrees, and so we cannot apply the Second Mapping Method . More precisely, for every i there are less than vertices having at least (1/2 + ,3)ti neighbors in i; We look for a weakly remappable subtree, that is, a good subtree Tj for which there are at least ,20 n vertices in Q each having at least (1/2 - ,4)tj neighbors in t; Assume that Tj is a weakly remappable subtree, and apply the second version of the Cleaning Lemma. We get a bipartite graph F'(A' , B') where A' c V(Tj), B' c B c Q, IBI = tJ, and every a E A' has at least t] neighbors in B' and every b E B' has at least (1/2 - 2,4)tj neighbors in A'. If the sizes of the color classes of Tj differ by at least ,3tj, then Tj C F' . Moreover, after embedding Tj into F' we would get a leftover dense subgraph

t;

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Tight Bounds for Embedding Bounded Degree Trees

of F' in which every b has at least ,3tj /2 neighbors in the leftover of A' . This dense subgraph can then be used to build T further. Hence, if Tj is weakly remappable but we cannot proceed the way we described above, then the sizes of its color classes differ by at most ,3tj. Another important observation is that if Tj is weakly remappable, then we can embed most of it into P', a subtree Tj of size at most 2K,4tj will be left out. Assume that after embedding Tj - Tj into P' the leftover of B' contains at least tJ /2 vertices, each having at least ,3tj neighbors in the leftover of A' . Then we are able to embed ij into the leftover by the help of the Main Mapping Procedure. Moreover, since v(Tj) ~ 2K,4tj, we can build T further by mapping a large subtree of it of size at least ,3tj /4. From now on we assume that none of the above conditions are satisfied, that is, the color classes of the weakly remappable subtree Tj are roughly of equal size, and apart from a few vertices, the neighbors of the vertices of B' are concentrated in a subset of A' of size (1/2 - 2,4 - ,3)tj . Set q =

IQI. We show the

following:

Lemma 12. If Q does not have a subset Q' such that the induced subgraph on Q' has minimum degree at least ,20 n , then Q has at most ,IO q vertices which have more than ,IOn neighbors in Q. Proof. Applying Lemma 3, the number of edges inside Q cannot be more than q,20n. An easy calculation shows that the number of those vertices of Q which have at least ,lOn neighbors in Q cannot be larger than 2,lOq .

• We claim that most of the subtrees are weakly remappable if none of them are remappable. More precisely: Lemma 13. In the above setup if none of the good Tj subtrees are remappable then at least (1 - ,/2)n vertices are in weakly remappable good subtrees.

,5

Proof. First observe that most vertices in Q have at most n /3 neighbors which are not in the union of the i; This follows from Lemma 12 and the fact that there are at most ,IOn vertices which do not belong to good subtrees, ID 1 U D2 U MI < 3,8 n and Tb has size at most ,5 n/ K 2 . Set q" = IQ"I where Q" is the subset of Q containing those vertices having at most ,1On neighbors in Q.

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Assume that Tj is not a weakly remappable subtree. Then there are at most ,20n vertices in Q" which have more than (1/2-,4)tj neighbors in Tj , with the remaining q" - ,20n vertices having fewer. Similarly, as i; is not remappable, at most ,20n vertices have more than (1/2 + ,3)ti neighbors in i; On the other hand, the number of edges going between Q" and the union of the good subtrees is at least q"n(1/2 - ,5/3). Putting these together we get the following inequality:

(,20 n + (q" - ,20n ) (1/2 -'IA))e + (,20 n + (q" _ ,20n ) (1/2 + ,3) )(n - c) 2 q"n( 1/2 _,5/3), where e denotes the number of vertices in subtrees which are not weakly remappable. The dominating term on the left is -q" e, while the dominating term on the right is -q"n,5/3. Therefore e < ,n/2, which implies the statement of the lemma. _

,4

Without loss of generality let {T1,1'2,...,Tk} be the set of weakly remappable good subtrees. By the second version of the Cleaning Lemma there is a partial remapping of i: which leaves a subtree of size at most 2K,4ti unmapped. Let I4. C I(Ti ) denote the vertices in the image of i; that are not used in the remapping. Let ri = 1I4.1. Observe that ri is roughly half of ti. The plan again is to gain a large dense subgraph in Q, which we use to build T further. Let us assume that we cannot remap any of 1'1, 1'2, .. .,Tk such that a dense subgraph is left in Q. Then (I4., Q) is sparse for every i, and L:ri ~ n(1/2 - 2,/3). 2 Consider the set UI4.. It has many edges, e(UI4.) ~ / 3, since G is non-extremal. We will show that there is a pair (I4., Rj) which has many edges and with I4. and Rj roughly the same size.

,n

Lemma 14. There exists i, j such that e(I4.,Rj) 21rirj, moreoverri ~ rj

implies rn/4

~

rio

Proof. Let us assume that, on the contrary, every pair has density less than v. Then

e(UI4.) ~

,/4 L

iii

rirj +

L e(I4.). i

The sum L:i e(I4.) so we concentrate on the first sum, which is at most ,/4L:iri(n-ri) . This last expression is at most ,n2/4-o(n2 ) . This contradiction implies that we can find a dense pair. = o(n2 ) ,

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Tight Bounds for Embedding Bounded Degree Trees

Assume now that all of the dense pairs are comprised of two sets with largely different sizes. Fix R; and consider the pairs (~, Rj) for which ri < rj - by assumption rj > 4rd, if (~, Rj) is a dense pair. The number of edges in the dense pairs containing R; is less than ,rd4 L rj ::; ,rin/8. Summing over every i we find that strictly less than 2 /3 edges are in pairs comprised of sets of largely different sizes. This proves the lemma. •

,n

We are now in a position to describe the re-mapping. If most of the subtrees are weakly remappable and none of them is remappable, then remap i; Tj for which (~, Rj) is a dense pair as in Lemma 14. Assume that ti ::; tj ' First remap most of i; a subtree of size at most 2K,4ti will be left unmapped. Since Tj is weakly remappable, we can find a bipartite graph F(A', B') according to the second version of the Cleaning Lemma. When remapping most of t; we may use vertices from B' . Notice, however, that apart from at most tJ/2 vertices, the vertices of B' have at most ,3tj neighbors in Rj . Moreover, every a E A' has at least tJ neighbors in B' . Delete those vertices from B' which are used to remap Tj together with those which have more than , 3tj neighbors in Rj . We lose at most tJ/2 + ti vertices and so B' still has at least 2tJ/5 vertices. Delete every edge going between R j and B', so that every vertex of B' has at least (1/2 - 2,3)tj neighbors in A' . We can then remap most of Tj such that a subtree of size at most 2K,3tj is left out . In addition, we do not use any vertex of Rj during the remapping. As (~ , Rj) is a dense pair, there are at least ,rirj/2 edges in between them. By Lemma 3 we can find a subgraph of this dense pair in which the minimum degree is at least,/2 r:~rj ' This is much larger than 2K,4ti or

2K,3tj, since r., rj and hence ti and tj does not differ by much. Therefore

we can map the unmapped subtrees of i; and t; and still have most of the edges of the dense pair left to map a new, so far unmapped subtree of T . As usual, we ensure that the mapping is properly connected through the use of the random subset M by the help of the Main Mapping Procedure. Observe that since the sets of dense pairs have sizes at least ,-20/2 we embed large new subtrees. 2.2.8. Description of the Embedding Algorithm. We are now ready to give the embedding method in the case where G is non-extremal and T has a broad subtree. After we are done with the mapping of the vertices of

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Tb - S, we apply the decomposition method of Section 2.2.2. Then we map the good subtrees of the decomposition in a top-down fashion. Starting from the root T , we map large subtrees ofT one-by-one. During the embedding the already mapped part of T will be a tree T' c T, that is, it will be connected . We proceed in the following way. We pick a leaf x of T' such that there is a subtree Tx c T(x) of size ")'-12 < v(Tx ) < ")'3hl ' We want to extend I for T' U Tx . As before, let Q denote the set of uncovered vertices of W - (MUD 1 UD2 ) . If Q has a large dense subgraph, then T x can be mapped by the Main Mapping Procedure, recall Remark 3. If not, then we will rethink the way we mapped certain subtrees, and look for remappable subtrees. If Q is not too small there will always be a remappable subtree of size at least ")'-20 . Remapping this subtree returns a large dense bipartite subgraph to Q which can be used to map Tx . If there is no remappable subtree then most of the already embedded subtrees are weakly remappable by Lemma 13. There exist two good subtrees which, upon remapping, return a large dense pair to Q (Lemma 14). This dense pair can be used to map T x . We proceed in this way until the size of Q drops below ")'lOn. Then we apply the matching method of Section 2.2.4, and find an embedding of T into G. This finishes the proof in the first non-extremal case. _

2.3. The second non-extremal case: T has a long subtree

In this section we discuss the case of embedding T when there is a subtree 7l c T with at least ,,),5 n / K3 vertices and at most ,,?n leaves. We first give a sketch of the procedure. A line in a tree T is a path P such that every vertex on P apart from possibly the endpoints has degree 2 in T . In this case, Ti must contain several long lines. We find a set of 2")'10 n lines, each of length 1/(5,,),2). We will cut out the midpoint from each of these lines, set aside the cut-out vertex, and glue the two new endpoints together. The result is a tree Ton (1 - 2")'1O)n vertices. We prepare G for the embedding ofT, beginning with finding the vertices to be covered by the long lines by the help of a randomized procedure. The~ we apply the embedding method of the previous section for embedding T into G. Since T is smaller than G, the embedding goes easily, even with

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Tight Bounds for Embedding Bounded Degree Trees

the restriction that the long lines of T are mapped onto the pre-determined long paths of G. Finally we address the cut-out vertices. We will extend the embedding of the long lines by finding a perfect matching in an appropriately defined bipartite graph in G. With high probability, this procedure finds an embedding of T into G. 2.3.1. Preparations for the embedding Finding random paths in G. In the first step we pick two random subsets

,B

M I , M2 C W (G). MI will have size ,lOn, while M2 will contain n vertices. Then we find the paths U, by the following randomized procedure: Set k = 2,9n.

Pick randomly and independently 2;"(2 edges from

G - MI - M2 with replacement. From these, form k sets of size 1/(20,2). We can connect the edges of each subset into a path of length by way of the paths of length three guaranteed by Lemma 8. Denote the paths by UI , U2 , ... , Ui: We discard those paths containing repeated vertices , where a vertex is repeated if either it appears twice in some U, or it is contained by u, and Uj for i i- j.

s:?-

Let v be any vertex in G. Let X; denote the number of occurrences of v in the randomly chosen edges, then EXv < This number follows a Poisson distribution,

,7.

That is, on average we expect to have less than ,13 n repeated vertices. The probability that there are more than 1O,13 n repeated vertices is by Markov's inequality at most 1/10. We call a path bad if either it contains a vertex at least twice, or it contains a vertex which appears in another path. In the latter case we call only one of those a bad path. After discarding the bad paths, the vast majority of the paths remain . With probability at least 90% we have at most < 2,lln bad paths.

logr;;n

A vertex v can be inserted into a path U, if v has two consecutive neighbors on the path. We also say that v is adjacent to Ui. Let us estimate the probability that v is not adjacent to a given path Ui. There are at least 2 edges in the neighborhood of v , since G is non-extremal. The probability

,n

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B. Csaba, 1. Levitt, J. Nagy-Gyorgy and E. Szemeredi

that none of the 20~2 randomly chosen edges is in the neighborhood of v is at most 1 1 (1 - ,) 20')'2 ::; e- 20')' < ,20. It is easy to upper bound the number of those paths for which there are less than (1 -,l1)n adjacent vertices. By Markov's inequality, the probability that one cannot insert more than (1-,l1)n vertices into a path is Using Markov's inequality again the number of such paths is at most ,lOn with probability ~ 1- ,/3. Hence, one can find k' = n paths such that these do not contain repeated vertices, and one can insert at least (1 )n vertices into any of them.

,9.

,9

,11

Finding long lines in T. First we show that if F is a tree having only a few leaves then it contains many long lines.

Lemma 15. Let F be a tree on t vertices with ct leaves for some 0 < c « 1. Then it is possible to find s = ct vertex disjoint lines PI, . .. .p; in F such that Ipil = 1/(4c) .

Proof. Choose a root p. Substitute every maximal line in F by one edge (but we keep p as is). Then every vertex in the new tree F' will have degree 1 or at least 3 except possibly the root. Since in a tree the number of leaves is bounded by the number of vertices with degree at least 3 plus 2, we have at most 2ct + 1 vertices and 2ct edges in P' . There were t - 1 edges in P, hence an average edge of F' corresponds to a line of length a least 1/(2c) . Cut out a part of this line of length 1/(4c) and glue together the resulting two endpoints. Repeat this procedure: construct a new P' as before, and then find a long path again by an averaging argument. We can continue this way, and find many paths of length 1/(4c), as long as the leftover number of edges in F is larger than t/2. • Apply the decomposition of Section 2.2.2 for Ti and for the rest of T separately. Call a good subtree of Ti long if it has size t and has at most 2,2t/ K 3 leaves. It is easy to see that almost half of the vertices of T[ are in long good subtrees. Apply Lemma 15 to find k' lines oflength 1/(4,2). We return edges of the lines to T as necessary in order to ensure that any two are at a distance of at least three from each other and that they are exactly 1/(5,2) long. We denote the resulting lines by PI, P2,.'" Pk'. Notice, that we may use up more than 40% of a long good subtree when determining the long lines. Therefore the long subtrees we use for the long lines will not contain more than 3,7 n vertices overall.

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Tight Bounds for Embedding Bounded Degree Trees

Let P i be a long line. Let Yi be the midpoint of P i and Xi and Zi its neighbors. Remove Yi from the path, and include the XiZi edge. This yields a new line P i and the cut-out vertex Yi. We repeat this for every 1 :::; i :::; k' , after which we arrive at a tree T which has (1 - k')n vertices. 2.3.2. Embedding T into G. Our next goal is to embed T into G. The decomposition of T carries over to one for T, comprised of good subtrees plus a small leftover. We use the embedding method of the previous case with minor modifications . In this case there is no use of D I and D 2 as their role is replaced by the random embedding of the long lines and the removal of one vertex from each line, which we embed at the end. The lines Pi are embedded first into the randomly constructed paths in G arbitrarily. The rest of the embedding of the decomposed T proceeds very similarly as before, with the exception that we begin with embedding TI, tEe long subtree, and when mapping a good subtree of T" containing the line Pi we apply the Main Mapping Procedure with the random set M 2 , i.e. connecting through M 2 , in order to connect the already embedded line to the rest of the subtree. Since overall we have , 9n such lines, and we use at most 6K 2 vertices to connect the embedded line, we do not use the vertices of M2 too quickly. After we are done with T", we put back the uncovered vertices of M2 into the set of vacant vertices of W - MI . Note that from this point the embedding procedure of the previous section is able to proceed as long as there are enough vertices in G that are uncovered. As we have retained k' » IMI I vertices to embed at the very end , this condition is always maintained. 2.3.3. Finishing the embedding. Recall that the probability that a vertex v cannot be inserted into any of the 20\2 randomly chosen edges is at most I I (1- ,)20/,2 ~ e-20/,. Since we choose the edges and then form the edge sets randomly, by CherI

noff's inequality every vertex v can be inserted to at least (1 - e- 20 /, ),9 n o(n) paths with probability 2: 1-1/n2 . After embedding T, we have the uncovered A c W(G) such that IAI = k'. If we can insert exactly one vertex of A to each of the U, paths, then we are done with the embedding of T into G. For showing that these insertions are possible we will apply the Konig-Hall marriage theorem. Construct a bipartite graph F(A, B) where A is as above and the vertices of B correspond to the randomly chosen U, paths. We connect a E A and

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b E B if the corresponding vertex can be inserted to the the path of b. We have n vertices in A and B . Recall that we discarded those paths into which less than (1 - ,11)n vertices could be inserted. Furthermore, with high probability every vertex in A is adjacent to most of the paths, thus, the minimum degree in F is large with high probability. Hence, the KonigHall conditions are satisfied with high probability. Therefore, we can find a perfect matching in F . This finishes the proof of the second non-extremal case. _

,9

3.

THE EXTREMAL CASE

We demonstrate how to embed a tree T into an extremal graph G. G is extremal if it is very close to being either a complete bipartite graph on color classes having size n/2, or it is the union of two complete graphs on n/2 vertices each. The two cases, while very similar, are handled separately. Note that that we did not use the full strength of the minimum degree of G in the non-extremal case. Even a minimum degree of (1/2 - c)n for some small e is sufficient for the arguments in the non-extremal case. The degree condition is critical, however, in the extremal case. If T has unequal color classes and G = KLn/2J.rn/21' then T ct G. If G = K n / 2 U K n / 2 , then it is not even connected. The First Extremal Case is when G is close to KLn/2J,rn/21, the Second Extremal Case is when G is close to K n / 2 U K n / 2 . We assume that 8(G) ~ n/2 + CK log2 n, where CK is a constant depending only on K (for more we refer to Lemma 17 and Lemma 18 below). In both extremal cases, the proof relies on this minimum degree. In the last section we will show that 8(G) - n/2 = !1(logn) in order to contain every bounded degree spanning tree. We will make use of a folklore result concerning trees in both extremal cases.

Proposition 16. Let J be any tree on m vertices. Then J has a vertex x E V(J) such that it is possible to group the vertices of J - x into two forests, J 1 and h such that m/3 ::; v(Jd,v(J2 ) ( ::; 2m/3) and there is no edge connecting J1 and h in J - x . The vertex x will be called a split vertex. Observe, that Proposition 16 applies for forests as well. If J is a forest, then turn it into a tree by adding

Tight Bounds for Embedding Bounded Degree Trees

edges to it. One can apply the proposition, find x , J1 and delete the recently added edges.

117

h, and then

In one of the extreme cases we have to cut a tree into two sets of equal size such that there is only a few edges going in between the two sets. Lemma 17. Let J be a tree on n vertices for some even integer n . J can be divided into two sets of vertex disjoint subtrees -li , h ,·· . ,Jt and Jt+i,'" .L, such that the number of vertices in each sets is n/2, and 8 ::; 210g 2 n . If the maximum degree in J is D, then the number of edges between the two set of subtrees is at most 2D log2 n.

Proof. Let A and B be two empty sets. Each time we split a subforest of J, we get a smaller forest and a larger forest. We continue to split on the larger forest until it can fit into A , then repeat the process on the smaller forest from the last splitting. The process is iterated until we fill A perfectly. After O(logn) splitting steps we get that IAI = IBI = n/2. We define the ai numbers for 1 ::; i ::; O(logn), ai denotes the empty space left in A at the ith step. In particular, ao = n/2. In the first step we apply Proposition 16 and remove the vertex x, splitting J into Jin and J out such that n/3 ::; V(Jin) ::; v(Jout). We put Jin into A, and set a1 = aO-V(Jin) from which it follows that a1 ::; ao/2. Define i1 so that

Define JO = Jout, and form a sequence of forests J O, J 1, ... J S l by recursively splitting the larger forest from the previous iteration, i.e. JI is the larger forest arising from splitting JI-1. Define 81 such that V(]51) < a1 < V(J s l - 1) and let ]51- 1 = Jt; + J~~t + x, where x is the splitting vertex chosen from ]51 -1 and where v (J~~t) < V(Jis.; ). Since after two applications of splitting the size of the larger forest is less than half of that of the original, we have that 81 ::; 2i1. We put the vertices of Jis.; into A , let a2 = a1-v(Jis.;). By the above conditions a2 ::; ad2 and a2 ::; v(J~~t) . If equality holds in the latter inequality, put the vertices of J~~t into A and stop. Let J S l = J~~t. This is the first step of the process. In general, assume that we have performed k - 1 steps and that there are ak vertices left to be placed in A, where

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for some positive integer ik . Let Sk-l = 81 + 82 +... 8k-l, and assume that 8t ~ 2(i t -it-I) for 2 ~ t < k. The forest J 8 k - l is to be split further, where ak < v(J8k- 1 ) < n/2 i k - 1 - 1 . Splitting on the larger subforest as above 8k have J 8 k , where Sk = Sk-l + 8k and

Define

8k J 8k- 1 = J,In8k + Jout

~ 2(ik - ik-d

times we

+x

where x is the splitting vertex chosen from J 8 k -

1

and

We put the vertices of Ji~k into A, let ak+1 = ak -v ( Ji~k) and let J 8k = J~~t. It is easy to see that ak+l ~ ak/2 and ak+l ~ v( J8 k ) . If equality holds in the latter inequality, put the vertices of J8 k into A and stop. Clearly, since at every step we halve the space left to fill in A, we need at most log2 n steps in order to get that IAI = n/2. The vertices of those forests which were not put into A are put into B. Since

s, ::; 2il +

L

2(it - it-I) = 2ik-l

l
In the other extremal case we have to find an almost proper 2-coloring of the vertices of a tree on n vertices, such that there are at most o (log n) edges which connect two vertices with the same color. Lemma 18. Let J be a tree on n vertices, n being an even integer. J can be divided into vertex disjoint subtrees J 1 , h ,... , J s plus at most 2log 2 n split vertices, such th at there is a proper 2-coloring of the vertices of the subtrees with equal sized color classes.

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Tight Bounds for Embedding Bounded Degree Trees

Proof. As in the proof of Lemma 17 our main tool will be Proposition 16. The goal is to give an "almost proper 2-coloration" of the tree J. The color classes will be of the same size, and apart from O(logn) edges, every edge will connect vertices with different colors. For a tree F let us define the coloring discrepancy, d(F) to be the difference of the sizes of the color classes. We denote by Xi, i = 1,2 the size of color class i, so that d(F) = Xl (F) - X2(F ) If F has m vertices then d(F) < m. If we have l trees , F I , F2, ... , F[ , each having size at most m , then the forest UFi has a proper 2-coloration such that d( Ui Fi ) < m.

I

I.

We exhibit a coloring algorithm which colors every vertex of J in O(logn) steps. We assume that there is a proper 2-coloring of J in the beginning, and the algorithm re-colors the vertices such that the color classes are of equal size and at most O(log n) edges connect vertices of the same color. Let ai be the number of vertices whose color has been reassigned to the 1st color class in the ith step, and let /3i be the number that have been reassigned to the 2nd color cl~s . Let ai = n/2 - ai and /3i = n/2 - $i . Before any recoloring, ao = /30 = 0 and ao = /30 = n/2. Throughout we will assume that ai ~ /3i , swapping the two colors at the end of the i - 1st step if necessary. We let dfn = /3i - ai ~ 0, representing the coloring discrepancy of the re-colored vertices of J. At the it h step there is a forest containing those vertices which have not been re-colored, the "outside forest". We denote its coloring discrepancy by d~ut , determined by the original 2-coloring of J restricted to Note that ai + /3i = v(J~ut). If is not a tree , then one can color it in many different ways.

n: n:

n;

We will maintain the following two conditions throughout the algorithm: C1: in the ith step v( J~ut) ~ n/2 i and

These conditions ensure that dtn is decreasing fast , since dtn ~ ~ut ~ v(J~ut) ~ «r». Therefore, after O(logn) steps we can get the desired coloring of J. We demonstrate how to achieve C1 and C2 in the first step and how to maintain these conditions throughout the procedure, implying the lemma. To begin, we apply Proposition 16 twice. After the first application we get two forests , and we apply splitting once more on the larger forest. One of the split vertices are put into the 1st color class, the other to the 2nd color

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B. Csaba, 1. Levitt, J. Nagy-Gyorgy and E. Szemetedi

class. We will have the three sub-forests J1, J2 and J3 such that v(Jd ~ n/2 for i = 1,2,3. Assume that d(J1) ~ d(h) ~ d(h), where the discrepancy is taken on the original coloring of J restricted to these sub-forests. Then we recolor J1 and J2 such that d(J1 U J2) ~ d(J3)' This is easily accomplished by swapping the colors in J1, say, so that the discrepancy of the recolored union is d(h) - d(J1) < d(h). We let dln = d(J2) - d(J1) and J~ut = h. Conditions C1 and C2 are clearly satisfied. Now assume that we have completed the ith step, v(J~ut) ~ n/2 i and dIn ~ d~ut· As before we apply Proposition 16 twice. First we split into two forests, then split the larger one, yielding the forests J1, hand J3. One of the split vertices are put into the 1st color class, the other to the 2nd color class. We have that v( Jt ) ~ n/2 i +1 for t = 1,2,3. The sizes of the color classes of Jt will be denoted by at and bt , where at ~ bt for t = 1,2,3. Observe that 2:(at + bd = ai + f3i·

r:

Claim 19.

2:199 at ~ ai·

Proof (of the claim). Suppose that

2: at > ai, then 2: b; < f3i.

Therefore,

contradicting condition C2. _ A forest Jt does not fit into the leftover, i.e., it cannot be re-colored without further splitting, if either at > ai or bt > f3i. We have just eliminated the first option . We say that Jt cannot be re-colored if bt > f3i.

Claim 20. There is at most one forest among J1, hand h which cannot be re-colored. Proof (of the claim). Assume that there are two forests which cannot be re-colored, say, J1 and h. Then b1, b2 > f3i' hence,

a contradiction. _ Assume that b1, b2, b3 ~ f3i' that is, each of the three forests can be re-colored. Further assume that d(J1) ~ d(h) ~ d(h). Then we first exchange the colors on the vertices of J1 so that the new coloring discrepancy is d(Jd - dIn easily seen to be at most d(h) + d(h). Next we exchange

I

I'

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Tight Bounds for Embedding Bounded Degree Trees

the colors of the vertices of h, which changes the coloring discrepancy to ~:l = Id(h) - d(Jl) - dIn d(J3)' We let J~~tl = h so that v(J~~tl) ::; n/2i+l. Hence, conditions C1 and C2 are satisfied.

I

II ::;

Now assume that b3 > {3i and bl , b2 ::; (3i. We claim that in this case d(J3) > d(Jl), d(h). If this were not true, then say d(Jl) > d(J3). Clearly

and so implying that

2{3i - b3 - 2al 2:: a3 + bl

-

al·

From the assumption that d(Jl) > d(h) we get that b3 < a3 implying that This implies that b3

+ bl

-

aI,

< (3i - aI, which is a contradiction.

In short, either all the three forests can be re-colored, or the one which cannot has the largest coloring discrepancy. In the latter case we can repeat what we did in the first step by first re-coloring J l, then J2. We i+1 +1 2:: ~n+1 and v(Jout i+1 . let Jout = J3. As above, we have ~ut ) ::; n/2i+1 . Agam, conditions C1 and C2 are satisfied. In at most log2 n steps we can find the desired 2-coloring. Since at every step we apply Proposition 16 twice, the total number of split vertices is at most 2log 2 n, and this finishes the proof of the lemma. _ Remark 5. As in the previous case, one can think of the above splitting procedure as giving labels to vertices of T. If x E V(T) then ..\(x) is either A or B, that is, the label of a vertex is a set. The number of those xy edges for which ..\(x) = ..\(y) is O(logn), since in such a case either x or y has to be a split vertex.

The following embedding lemma will be applied in both type of extremal cases: Lemma 21. Let F be a tree with constant maximum degree K and color classes QF and RF. Let H = (Q H,RH) be a bipartite graph such that 1. q = IQFI = IQHI and r = IRFI = IRHI and m = q + r;

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B. Csebe, 1. Levitt, J. Nagy-Gyorgy and E. Szemeredi

2. if u E QH then deg (u) 2

(1- 3k )r;

3. if v E RH then deg(v) 2 (1-

3k )q;

4. Q C QF and R C RF are such that

IQI + IRI =

O(logm),

where Q and R denote the set of those vertices of F which are a priori restricted to be mapped onto given vertices of QH, respectively RH. Assume further that eF(Q, R) = O. Then F is a spanning tree of H ifm is sufliciently large. Proof. We will find a bijective, adjacency preserving mapping ¢ from V(F) to V(H) . Note that ¢(Q) and ¢(R) are already given. First we map the vertices of NF(Q) . If x E NF(Q) then we pick a vacant vertex v E RH-¢(R) such that if Y E NF(X) n Q then v is adjacent to ¢(y). We have at least 2r /3 - O(log m) vertices to choose from since the minimum degree is large in H and we have to map O(logm) vertices. Then we let ¢(x) = v. Let R = R U NF(Q). Clearly, IRI = O(logm) since t:.(F) is a constant. We extend the definition of ¢, beginning with a random bijective mapping ¢' : QF - Q -7 QH - ¢(Q), mapping every x E QF onto ¢'(x) E QH. Let ¢(x) = ¢'(x) for every x E QF - Q. We estimate the probability that the vertices of NF(R) are mapped onto neighbors of ¢(R). We call ¢ good for R if x E NF(R) and Y E NF(X) n R implies ¢(x)¢(y) E E(H). Let x E NF(R) and assume that Yl, Y2, .. . ,Yt E R are adjacent to x, where 1 ~ t ~ K . By the minimum degree condition of H we get that I ni N(Yi)1 2 2q/3 - O(logm), and so the probability of choosing a good ¢'(x) from the uncovered vertices in QH is at least 1/2 for every x E NF(R) . This implies that the probability that ¢ is good for R is at least 2- K1R1= 1/ poly (m), where poly (m) is some polynomial of m . We extend ¢ to an embedding of F by a matching argument. Let R~ = RF - R and R~ = RH - ¢(R). Construct the auxiliary bipartite graph L(R~, R~), connecting u E R~ to x E R~ iff u is adjacent to ¢(Yl) ,¢(Y2) , ... ,¢(YK) in H, where Yl,Y2 , ... ,YK E QF are the neighbors of x . If we can find a perfect matching in L then we have extended ¢ to an embedding of F into H . For proving the existence of the perfect matching we check the KonigHall conditions. In particular, we show that every u E R~ is adjacent to at least r /2 vertices in R~ and that every x E R~ is adjacent to at least r /2 vertices in R~ with high probability.

123

Tight Bounds for Embedding Bounded Degree Trees

Let least

U

E R'll" The probability that ux E E(L) for some x E R~ is at

((l-k~ )q)K!(q - K)! ---=-=----q!---

~

(

1)K

1 - -2K-

> 0.56,

as this is the probability that the neighbors of x are all mapped to NH(u). By Azuma 's inequality (see e.g., in [1]) we get that every u E R His adjacent to more than r /2 vertices of R~ with very high probability. In fact the probability that the degree of u in L is less than r /2 is at most 2e- cm for some c > O. Hence, the probability that ¢ is good for R and every vertex of R Hhas at least r/2 neighbors in L is ~ (1- 2e- cm ) - (1-I/poly (m)) > 1/ ( 2 poly (m)) if m is sufficiently large. Next we show that every x E R~ has at least r/2 neighbors in R H. Let the neighbors of x E RF be Yl, Y2, ... ,YK E QF. By the minimum degree condition in H we have that ¢(Yl), ¢(Y2), .. . ,¢(YK) has at least 2r/3 - O(logm) > r/2 common neighbors in R H. Indeed, this holds for every ¢. This proves the lemma. _

Sketch of the embedding algorithm. Denote by SP the induced subforest of T which contains the split vertices and those vertices of T which are at a distance at most four from the split vertices. Clearly, v(SP) ::; 2K4 log2 n. SP may contain many connected components. In both extremal cases our first goal is to embed the forest SPin such a way that those vertices of SP which are adjacent to some vertex in V(T) - V(SP) are mapped onto "large degree vertices" of G. In the first extreme case the large degree vertices are adjacent to at least n( 1/21/(4K)) vertices inside the set to which they belong (either in A or in B). In the second extreme case large degree vertices are those with at least n(1/2 -1/(4K)) neighbors in the other set. Since v(SP) = O(1ogn), after some preparations we can apply Lemma 21 in order to map the vertices of T-SP.

3.1. The First Extremal Case

,n

In this case W(G) is partitioned into 2. and e( GIA) ,e( GIB) ~ (n~2) -

A and B such that IAI

=

IBI

= n/2 ,

First we state a simple lemma on the degrees of the vertices of A and B, we leave the proof to the reader.

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B. Csebe , 1. Levitt, J. Nagy-Gyorgy and E. Szemeredi

Lemma 22. Let 0 < n < 1/2. Then the number of vertices of A having less than n/2 - nn neighbors in A is at most ~ . The analogous for B also holds. • Preparations for the embedding - Step 1: Decomposition of T. By Lemma 17 we find a bipartition of the vertex set of T into two sets , X and Y such that IXI = WI = n/2 and e(X, Y) = O(logn) . Recall that SP is the sub-forest of T which contains the split vertices and those vertices of T that are at a distance of at most four from the split vertices. Let SPI , SP2 , ••• , SPt denote the components of SP. Since SP is an induced subgraph, there are no edges connecting SPi and SPj for i =f. j. For every 1 ~ i ~ t we pick an arbitrary split vertex belonging to SPi to be the root of SPi . Preparations for the embedding - Step 2: Switching and finding stars. We define the inner degree of v E A as Nc(v) n AI and the outer degree of v E A as Nc(v) n BI. The inner and outer degree of u E B is defined similarly.

I

I

Assume that u E A and v E B are such that

e(A - u + v) + e(B - v + u) > e(A) + e(B). Switch u to B and v to A, and look for another pair of vertices which can be switched. At every switching step the total number of edges inside A and B is increased , therefore this process will stop in a finite number of steps . Observe that if there exists u E A and v E B such that both have inner degrees of at most n/4, then it is still possible to perform a switching step. Hence, at the end of the switching procedure we have that every vertex in one of the parts, say A , has an inner degree of at least n/4. Observe, that after the switching even the sparser set will contain at least n 2 /4 - 2,n2 edges, so both sets will be 2,-extremal. There may be up to 16,n vertices in B which have at most n/4 neighbors in B (see Lemma 22). Denote the set of these vertices with low degree in B by B' . We determine IB'I vertex disjoint stars in B, each star centered on a vertex of B' connected to K vertices from B \ B'. We determine the stars greedily. For each u E B', assign K vertices from N (u) n (B \ B') as long they are not yet assigned to another vertex of B'. Call a vertex of B' marked if it is designated as the center of a K -st ar, and call a vertex in B \ B' vacant if it is not assigned to a marked vertex of B' .

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The greedy procedure stops when every unmarked vertex of B' has less than K vacant neighbors in B \ B'. If no unmarked vertex in B' has K vacant neighbors, then apart from at most 16K,n vacant vertices the rest of B - B' has no neighbor among the unmarked vertices of B'. Denote the set of unmarked vertices of B' by B~ . B has n/2 vertices, of which at most (K + 1)16,n belong to the stars and at most (K + 1)16,n vertices have a neighbor in B~. Each of the remaining at least n/2 - 32(K + l)rn vertices has at least CK logn neighbors in A. It follows that an average vertex of A has at least

CK

Iogn

n/2 - 32(K + l)rn n/2

>

K

neighbors among these vacant vertices. Hence, there is at least one w E A which has at least K vacant neighbors . We pick an arbitrary vertex v E B~, switch it with w, mark w, and assign the K vacant neighbors to w. Note that since the greedy algorithm could not assign K vacant neighbors to v, the outer degree of v before switching was large, at least n/2 -16(K + l)rn. The simple averaging argument ensures that there always will be vertex in A having K vacant neighbors, and so we can continue the process to guarantee that (1) every vertex of A has inner degree at least n/4, and either (2) the inner degree of a vertex u E B is at least n/4, or (3) u sits in the center of a K -star such that the leaves of this star have inner degree at least n/4.

Preparations for the embedding - Step 3: Embedding S P. It is easy to see that one can embed the induced sub-forest SP greedily if CK ~ 2K4 . Still, we have to be careful to ensure that certain vertices of SPare mapped onto vertices with large inner degree. Let x E V(SP), and denote its neighbors by Yl, Y2, ... , YK. If x is the vertex to be mapped and some of the YiS have already been mapped, then we must map x onto an uncovered vertex u E >.(x) such that uI(Yi) E E(G) for every mapped Yi. We begin mapping the components of SP with their roots. We obey the following rules when mapping a neighbor of an already mapped vertex: Rl. If x is mapped to a vertex with inner degree at least n/4, Yi is unmapped and >.(x) = >'(Yi) , then we map Yi onto an uncovered vertex which has inner degree ~ n(1-1/(4K)) /2.

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R2. If X is mapped to the center of a star, Yi is unmapped and '\(x) = '\(Yi) then Yi is mapped to an uncovered leaf of the star. R3. If x is mapped to a leaf of a star such that the center of the star is uncovered, Yi is unmapped and '\(x) = '\(Yi), then we map Yi onto the center of the star. R4. We never cover the center of a star by a vertex x if at least one leaf of the star is already covered by a vertex Y, for which xy ~ E(T). It is easy to satisfy R2 and R3. R4 can be satisfied in the course of the embedding of SP if 8(G)-n/2 ~ 2v(SP) . For Rl observe that by Lemma 22 at most 12K 2,n vertices have inner degree less than n ( 1 - 1/(4K) ) /2, and so out of the at least n/4 - v(SP) -12K2,n > n/5 choices one can always find an uncovered vertex for Yi with sufficiently large inner degree. In this way, every vertex of S P which is at a distance of at least four from every split vertex is mapped onto a vertex of G with large inner degree.

The distance four condition in the definition of SP is needed for the following reason. Assume x is a split vertex with, say, '\(x) = A, xy E E(SP) and '\(y) = B . When mapping Y to B, it is possible that it is mapped to a leaf of a star. When we apply R3 and R2, we find that only a vertex which is of distance four from x can be mapped onto a vertex of large inner degree. The problem we may face after finishing the embedding of S P is that some of the stars may have been damaged. That is, we may have covered some leaves of certain stars such that the center vertices are still uncovered. We solve this problem by finding new leaves for such center vertices. We accomplish this using the same method applied to find the stars initially. Say that there is an uncovered vertex u E B' such that some of the leaves of its star were covered. Assuming that CK logn ~ v(SP) + 2K, every vertex will have unmapped outer degree of at least 2K even after embedding SP. Hence, either we can find K vacant uncovered neighbors in B having inner degree at least n/4, or by averaging we will find an uncovered v E A such that v has at least K vacant uncovered neighbors, all having inner degree at least n/4. In summary, SP is embedded such that every vertex which is at a distance of at least four from every split vertex is mapped onto a vertex with inner degree at least n(1-1/(4K)) /2. Moreover, if x E V(T) - V(SP), Y E V(T), and xy E E(T) then '\(x) = '\(y). That is, every subtree in the leftover is labelled to be mapped entirely to A or to B .

Tight Bounds for Embedding Bounded Degree Trees

127

Embedding T - Step 1. First we state a simple but useful lemma. Lemma 23. Let u, v E B. Then there is a U - v-path of length at most five entirely in B. The path can be found even if we discard at most n/5 vertices from B , as long as the stars of U and v remain intact. An analogous statement holds in case u ,v E A. Proof. We construct a u - v-path of length five when u, v E B'. Let UI be an arbitrary leaf of the star centered at U and let VI be a leaf of the star centered at v. Since the inner degree of UI and VI each is at least n/4, even after deleting n/5 vertices both will have at least n/20 neighbors left in B. Out of these vertices in the neighborhood of UI we can find a vertex U2 with inner degree at least n/2 - 80,n (see Lemma 22). Therefore, U2 has a neighbor V2 E NC(VI) n B. The U- v-path of length five is: u, UI, U2, v2, VI, V.

If at least one of u ,v belong to B - B', then one can find an even shorter path, as is the case when both vertices belong to A. We leave the proof of these cases to the reader. •

We find two subtrees of T , denoted by TA and TB such that V(TA) C A and V(TB) C B , and n] K 7 < V(TA),V(TB) ~ n] K 5 . We embed these subtrees into A or B by covering every vertex with inner degree at most n(l - 1/(3K)) /2 , thereby preparing G for the application of the general tree embedding lemma, Lemma 21. Recall the decomposition of T from Lemma 17. After the first splitting, the smaller forest is assigned to A. It has size at least n/3, hence its largest subtree TI is of size at least n/(3K). Then we split the leftover, and the smaller forest is assigned to B. The size of it is at least n/6, hence, its largest subtree T2 has size at least n/(6K). We find TA C TI and TB C T2 . Finding TA and TB are very similar, we discuss finding TA. Since V(TI) > n/(3K), TA will be a proper subtree of TI. Let r E V(TI) be the unique vertex in V(TI ) which has a neighbor in SP (if there were two neighbors, we would get a cycle). Beginning at r, form a maximal path r = XO,XI,X2, .. ' , Xk such that TI(xd is the largest subtree of TI rooted at a child of Xi-I and such that v( TI (Xi)) > n.]K5 for every i ~ k. Choose Z such that TI( z) is the largest subtree among the children of Xk. We let TA = TI(Z) and let z be the root of TA. Clearly, n/K7 < V(TA) ~ n/K5 . In the same way we determine TB C T2 such that n]K7 < V(TB) ~ ti] K 5 . Observe, that TI - TA has only one vertex which is adjacent to TA, and similarly for TB. Since we put T2 into B after two splittings, there may

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B. Csaba , I. Levitt, J . Nagy-Gyorgy and E. Szetneredi

be two edges from TB to the rest of T, one connecting to T2 connecting to SP.

-

TB and one

We discuss the algorithm for embedding TA and TB. This time we give the details for TB, which is somewhat harder than the case of TA. We begin with putting the uncovered vertices of B in increasing order according to their inner degrees, writing Ul, U2 , . . . ,Ue, where n/2-v(SP) :S f < n/2. Thus we may refer to the first m vertices of B, by which we mean the set {u1, U2 , . . . ,Urn}. The goal is to cover as many vertices from the beginning of this ordered list as possible. We first map the root of TB onto an uncovered vertex which has inner degree ~ n(1-1/(4K)) /2. Since TB may have a vertex which is adjacent to S P, we are careful to map that vertex onto an uncovered vertex having inner degree ~ n(1-1/(4K)) /2 . A simple general rule will always be applied: when mapping x we map it on an uncovered vertex U such that UI(Yi) E E(G) for every mapped neighbor Yi. Apart from this obvious requirement we obey rules R2 and R3, as described during the embedding of SP. Say that in the course of the embedding the most recently mapped vertex is Xi onto Uj for some 1 :S i :S i , and that the first uncovered vertex of B according to the ordering is Ut, where i :S t. Our next goal is to cover u-. By Lemma 23 there is a path of length at most five between Uj and u-, the length of which is denoted by f. Consider the induced subtree containing unmapped vertices of TB(Xi), denoted T~(Xi), having depth f and being comprised of every vertex below Xi which is at distance at most f from Xi. This subtree has at most K 5 vertices, which we embed greedily, but in such a way that one of its leaves will cover Ut . By Lemma 23 this is possible to achieve. Then we pick the next vertex in the ordering of V(B), covering which may again embed at most K 5 vertices from TB. That is, by mapping at most K 5 vertices of the tree , we can cover at least one uncovered vertex of B' . Hence, after mapping at most K 5 1 B'I :S K 5 16')'n vertices of TB we will have no uncovered vertex left in B' . After covering every vertex of B' we will try to cover as many "small" inner degree vertices as possible. Again, we apply Lemma 23 and find that by mapping at most K 5 vertices of TB we can cover any uncovered vertex we want. An easy calculation shows that we can cover the first n/(2K 12 ) vertices of B in this way. By Lemma 22 every uncovered vertex left in B will have inner degree at least n/2 - 4K- 8n > n(1-1/(4K)) /2 .

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Tight Bounds for Embedding Bounded Degree Trees

Observe that when mapping the vertices of TA we get at least the same bounds for the inner degrees of uncovered vertices of A. Remark 6. After this step every uncovered vertex has inner degree at least n(1 -1/{4K)) /2. We covered at most n/K5 vertices of A and of B , hence the minimum degree inside these sets is at least n(1-1/{4K)) /2 - n/K5 . Embedding T - Step 2. It is now a simple task to finish the embedding of T by the help of Lemma 21. Denote the forest induced by unmapped vertices which have to be mapped into A by i; and let T denote the forest of unmapped vertices which have to be mapped into B .

i:

Take an arbitrary proper two coloring of letting Q and R denote the color classes, and set q = IQI and r = IRI. Divide the uncovered vertices of A randomly into two sets, QA and RA, where IQAI = q and IRAI = r . Using Azuma 's inequality and Remark 6 every u E QA will be adjacent to at least r( 1-1/{3K)) vertices in RA, and every vERA will be adjacent to at least q( 1 - 1/{3K)) vertices in QA. There are at most v{SP) + 1 vertices which are a priori restricted to be mapped onto given vertices of A - the vertices of SP and the unique vertex in T; adjacent to TA. Since the require~nts of Lemma 21 are satisfied, we conclude that f can be embedded . T is embedded similarly. Hence, we conclude that in the first extremal case T c G. • Remark 7. Note that case.

CK

= 4K4 is sufficiently large in the first extremal

3.2. The Second Extremal Case In this case G is close to K n / 2 ,n / 2 ' This case is in fact very similar to the previous one, so the emphasis will be on the differences. We discuss the switching procedure in detail, and give an outline of the rest of the embedding method.

,n

We assume that W{G) is partitioned into two vertex classes of size n/2, 2 . Observe that by the minimum A and B, such that e{A) + e{B) ::; degree condition, every vertex will have at least CK log n neighbors in both vertex classes. We need a very similar statement to Lemma 22:

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Lemma 24. Let 0 < TJ < 1/2. Then the number of vertices of A having less than n/2 - TJn neighbors in B is at most~. The analogous statement is true for B. •

Preparations for the embedding - Step 1: Decomposition of T. We start by finding a decomposition of the tree T by Lemma 18 into a set of subtrees T I , T2 ," " T; with the following properties: (1) s = O(logn) , (2) there is a proper red-blue coloring of these subtrees such that overall we color n/2 vertices red and n/2 vertices blue and (3) there are at most O(logn) edges of T inside the color classes, each of which is incident to a split vertex. As before, if x E V(T) then A(X) = A or B indicates into which set we map x. It is easy to see that the largest subtree of this decomposition, TI has at least n/(3K) vertices. Given the decomposition ofT, we define the forest SP as in the previous extremal case as the induced sub-forest of T containing every vertex being at a distance at most four from a split vertex. By the definition of SP, there are no two components of SP with an edge connecting them.

Preparations for the embedding - Step 2: Switching and finding stars. We repeatedly perform a switching procedure whenever it is possible. If we identify a vertex v E A and a vertex u E B such that e(A , B) < e(A - v + u, B - u + v), then we let A = A - v + u and B = B - u + v. Since the number of edges in between A and B increases at every step of the switching , this is a finite process. Note also that when we are finished with the switching , one cannot find a pair of vertices v E A , u E B such that N(v) n n/4 and N(u) n n/4. Therefore, in one of the vertex classes, say A , we find only vertices which have at least n/4 neighbors in the other vertex class, B . We will use the notion of the inner and outer degree, as defined in the previous extremal case.

I

BI ::;

I

AI ::;

We pay special attention to those vertices of B which have very small outer degree. Observe that we may have at most 8, n vertices in Beach having outer degree less than n/4. Denote their set by B'. As before, we find IB'I vertex disjoint stars on K + 1 vertices. The leaves of these stars come from A , and the center vertices are either from A or from B . We determine such stars as follows. Let A v be the set of vacant vertices, and set A v = A and A' = 0 at the outset. One-by-one we look for K vacant neighbors for every u E B'. We have the following cases.

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Tight Bounds Eor Embedding Bounded Degree Trees

Case 1: If u is adjacent to Vl , V2, .. . , VK E A v , then the new star will have center u and leaves Vl, V2 , ... , v K. We let A v = Av - {Vl, V2, .. . , v K }, and A' = A' + {Vl, V2, . . • , V K }. Case 2: If u does not have K vacant neighbors, but there is a star with center v E A v and leaves Vl, V2 , • . . , VK E A v , then we switch u with v. The new star will have center v and leaves Vl ,V2, .. . ,VK . We let A v = A v + u - {Vl , V2, ... , V K} and A' = A' + {Vl' V2, ... , V K } .

Lemma 25. Assume that there exists a u E B ' such that it does not have K vacant neighbors. Then A v contains a star on K + 1 vertices. Proof. Suppose we have found t stars. Clearly, t > log., by the minimum degree condition on G. Let us assume to the contrary that A v does not contain a star on K + 1 vertices. If u does not have K neighbors in A v then u has at most K t + t + K - 1 neighbors in A after the first switching procedure is concluded. At most K t of these neighbors are from A', at most K - 1 are from A v and we apply the switching of Case 2 at most t times. Since t > K, we have that Kt + t + K - 1 ::; (K + 2)t. The following claim is easy to prove:

Claim 26. If at the end of the first switching procedure B has a vertex with outer degree d, then no vertex of A has inner degree larger than 2d.

• This implies that the vertices of A' have inner degree at most 2(K + 2)t. Assuming A v has no star on K + 1 vertices, we have the following lower bound on the number of edges in between A and B :

e(A , B) ~

n2

"4 -

2K(K + 2)t2

n

-

(K -1)2"'

We also have the following upper bound on e(A, B) by inspecting the outer degrees of the vertices of B:

e(A, B) ::;

t~ +

(i - t) ;.

From these we get the inequality 2

0::; 2K(K + 2)t -

n 4n t + (K -1)2"'

This inequality is not satisfied for t E [2K, S,n] . •

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Hence, if we fail to apply the method of Case 1 then Case 2 can always be applied , and the stars can be determined.

Preparations for the embedding - Step 3: Embedding SP. Embedding S P is very similar to the previous case. Our goal is to map x onto an uncovered vertex U E ..\(x) such that if xy E E(T) and we have already mapped Y then uI(y) E E(G) . We begin mapping a component of SP by its root . As in the previous extremal case, we have four rules concerning the embedding of SP . Rl/ , If x is mapped to a vertex with outer degree at least n/4, Yi is unmapped and ..\(x) =I=- ..\(yd , then we map Yi onto an uncovered vertex which has outer degree 2:: n(1-1/(4K)) /2.

R2/. If x is mapped to the center of a star, Yi is unmapped and ..\(x) then we map Yi to an uncovered leaf of the star.

=I=-

..\(Yi)

R3/. If x is mapped to a leaf of a star such that the center of the star is uncovered, Yi is unmapped and ..\(x) =I=- "\(Yi) , then we map Yi onto the center of the star. R4/. We never cover a center of a star by a vertex x if at least one leaf of the star is already covered by a vertex Y for which xy tt E(T) . As before, it is easy to satisfy the above rules, and as before we may face the problem that after finishing the embedding of SP, some ofthe stars may have been damaged. That is, we may have covered some leaves of certain stars without covering their centers. We solve this problem by finding new leaves for such center vertices. Suppose that there is an uncovered vertex u E B' such that some of the leaves of its star were covered. Assuming that CK logn 2:: v(SP) +2K, every vertex will have an uncovered outer degree of at least 2K even after embedding S P. Hence, either we can find K vacant uncovered neighbors in B having outer degree at least n/4, or by averaging we will find an uncovered v E A such that v has at least K vacant uncovered neighbors, all having outer degree at least n/4, by Lemma 25. In this way we find the missing leaves for the damaged stars. In summary, after embedding SP every vertex which is of distance at least four from every split vertex is mapped onto a vertex which has outer degree at least n( 1-1/(4K)) /2. Moreover, if x E V(T)- V(SP), Y E V(T), and xy E E(T), then ..\(x) =I=- ..\(y). That is, ..\ represents a proper 2-coloring for every subtree in the leftover.

Tight Bounds for Embedding Bounded Degree Trees

133

Embedding T - Step 1. First we state a simple but useful lemma whose proof is very similar to the proof of Lemma 23. Lemma 27. Let u, v E B . Then there is au - v-path of length at most six, with vertices alternately belonging to B and A. The path can be found even after discarding at most n/5 vertices, as long as the stars of u and v remain intact. The analogous statement holds in case u, v E A. _ As in the previous extremal case, we find a subtree of T, denoted by T' such that n]K7 < v{T') ::; n] K 5. As in the previous extremal case, when embedding this subtree we cover every vertex with outer degree at most n(l- 1/{3K)) /2 , thereby preparing G for the application of the general tree embedding lemma , Lemma 21. Recall that the largest subtree in the decomposition, T I , has at least n/{3K) vertices. Again, we order the uncovered vertices of W(G) according to increasing outer degree and continue the embedding of T by that of T' . In order to cover the low outer degree vertices we will repeatedly apply Lemma 27, achieving at the end that we cover the first n/{2K I3 ) vertices in the list. We omit the details as they are very similar to the one discussed in the first extremal case. By Lemma 24 every uncovered vertex left in W(G) has outer degree at least n/2 - 4K-7n > n(1-1/{4K)) /2 .

Remark 8. After this step every uncovered vertex has outer degree at least n(1-1/{4K)) /2. We covered at most n/K5 vertices of W{G), and so the minimum degree in the leftover is at least n(1-1/{4K)) /2 - n/K5 . Embedding T - Step 2. In order to finish the embedding of T we observe that the conditions of Lemma 21 are satisfied for the leftover forest T, and so T can be embedded. Hence, we conclude the second extremal case. _ Remark 9. Notice, that as well.

CK

= 4K4 is sufficiently large in this extremal case

4. LOWER BOUND FOR THE MINIMUM DEGREE IN G In this section we show that 8{G) -n/2 = n(1ogn) for both extremal cases. First we need a definition.

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Definition 2. A balanced cut of a graph is a partitioning of its vertex sets into two subsets, such that their sizes differ by at most one. The size of a cut is the number of edges going between the two partition sets. As we will see, the minimum size of a balanced cut of T provides a lower bound when embedding it in the first extremal case. while, the maximum size cut provides a lower bound for the second extremal case. In both cases T is a complete ternary tree on n vertices, and the root r of T is the only vertex having degree three, all other vertices have degree 4 or 1. We define the height h(x) of x as the number of edges on an x - y path, where y is the closest leaf to x.

4.1. The First Extremal Case

Lemma 28. Let T be a complete ternary tree of height h. Then the size of any balanced cut of T is at least lh/2J. Proof. Let A and B be the partition sets of a minimum balanced cut of T , we assume that IAI 2: IBI. Let X : V(T) -+ {-1, 1} be a coloring function . We initialize the coloring X(x) = 1 for every x in V(T), then apply a color-swapping process such that at the end of the process A = X- 1 (1) and B = X- 1 (- 1). The Color-swapping Process: Initially the color of every vertex is 1. Traverse T by a depthfirst search. If there is a vertex x such that x and its ancestor are not in the same partition set then multiply the colors of the vertices of T(x) by -1. We call this operation the color-swap of T(x). At the end of the process we get the desired coloring X such that B = X- 1 (- 1). It is clear that this process will always yield a coloring which corresponds to the cut, and that the number of color-swaps performed is the size of the cut. For any coloring X, label every vertex x by the signed difference of the size of the color classes of T(x). More precisely, for every x let L(x) = 2: y E :l' (x ) X(y). In the procedure described, the labels are updated whenever a color-swap operation is performed. When we swap the colors of T(x), the label of a vertex y on the r - x path changes by adding -2L(x) and the labels of the vertices of T(x) are A

= X- 1 (1) and

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Tight Bounds for Embedding Bounded Degree Trees

multiplied by -1. It is useful to represent the labels in a modified ternary numeral system in which the digits are -1, 0, 1. Such a representation will be subscripted by a "3". For example, 101(-lh = 33 + 31 - 1 = 29. Every integer can be represented this way and this representation is unique. Let f.k = 11 . .. 13 where the subscript k refers to the number of Is. The following facts hold: We introduce the notation Iiall for an integer a, defined to be the number of non-zero digits in the modified ternary representation of a. It is easy to see (by induction on the number of digits of a) that if Ibl ::; lal then Iia ± bll ::; lIall + IIbll· We refer to Iiall as the norm of a. 1. Initially the label of any x E T is L(x) Hence, initially II L(x)11 = h(x) + 1.

= £k where k = h(x) + 1.

2. At the end of the process II L(r)11 ::; 1. 3. After a color-swap of T( x) the labels of the vertices on the r - x path increase or decrease by 100... O(-1 h. If x(x) = 1, then the labels

decrease, otherwise increase. Here the number of Os is h(x) . Initially II L(r)11 = h(r) + 1, and at every color-swap L(r) increases or decreases by a number which has norm two. If the size of the minimum balanced cut is m, then there are m color-swaps. Since one color-swap can decrease II L(r)l/ by at most two, at least lh/2J color-swaps are needed in 1 at the end. The lemma follows. • order for L(r)

I

I ::;

We define the extremal graph G = (W, Ea). Set h = log3 n, nl = n/2 - h/16 + 1 and n2 = h/16 - 1. The vertex set of G is decomposed into four disjoint subsets, W = Xl U X 2 U Yl U Y2, where Xl and X 2 are both nl-cliques. Let Yl and Y2 each be sets on n2 vertices and let Y = Yl U Y2 be an independent set. Finally, we form the complete bipartite graphs between (XI, Y) and (X2, Y). It is easy to see that 8(G) = nl -1 + 2n2 = n/2 + h/16 - 2. Assume there is an embedding I : V -t W of the complete ternary tree T into G. In order to apply Lemma 28, let A = I-l(Xl U Yl) and B = I- l(X2UY2). We have IAI = IBI = n/2, and therefore eT(A, B) ~ h/2. Since ea(XI, X 2 ) = 0, one endpoint of every edge going in between A and B is mapped to a vertex in Y . The maximum degree of T is 4, hence there are at most 41Y1 edges of T with one endpoint mapped onto some vertex in Y. Since 4/Y/ < h/2 we have arrived at a contradiction: T et G.

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4.2. The Second Extremal Case

Lemma 29. Let T be a complete ternary tree with height h. The size of any balanced cut of T is at most e(T) - lh/2J. Proof. Let A and B, IAI > IBI, be the vertex sets of a balanced maximum cut of T . We show that the number of those edges which have both endpoints belonging to the same set is at least lh/2Jin a manner similar to the previous case. We apply a color-swapping process similarly to the previous case, but with two modifications. First, the coloring function X : V(T) --+ {-2, 2} is initialized as a proper 2-coloring of T with X(r) = 2. Second, we multiply the vertices of T( x) by -1 when x and its ancestor are in the same partition set . After applying the Color-swapping Process, we have the following: 1. The labels of the vertices of T depend on their height and the height

of the tree: (a) If h(r) and h(x) are even, then L(x) = 1(-1)

(-lh .

= 11 13, (c) If h(r) is odd and h(x) is even, then L(x) = (-1)11 ... 13, (d) If h(r) and h(x) are odd, then L(x) = (-1)(-1) ... (-lh. (b) If h(r) is even and h(x) is odd, then L(x)

In each case the number of digits of L(x) depends on the parity of h(x). If h(x) is even, then the number of digits is h(x) + 2, otherwise it is h(x) + 1. 2. At the end of the process L(r) of h.

= 0 or 1(-1 h, depending on the parity

3. After the color-swap of T(x) the labels of the vertices on the r - x path increase or decrease. If h(x) is even, then the absolute value of the change is 10 ... 013, where the number of zeros is h(x). If h(x) is odd, then the labels change by 10 ... O( -lh, here the number of zeros is h(x). The above facts can be proved easily by induction. Application of the same reasoning as before implies the statement of the lemma. _

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Tight Bounds for Embedding Bounded Degree Trees

This time the extremal graph G = (W, Ea) is defined as follows. First, set h = log3 n, nl = n/2 - h/16 + 1 and n2 = h/16 - 1. The vertex set of G is decomposed into four disjoint subsets: W = Xl U X2 UYl U Y2 , where Xl and X 2 are two independent sets of size nl, and GIXIUX2 is isomorphic to Kn1 ,nl' Let Yl and Y2 be two sets on n2 vertices , and let Y = Yl UY2 be an independent set. Finally, let Glx1uy and Glx2uy be isomorphic to K n 1 ,2n 2 ' It is easy to see that 8(G) = nl + 2n2 = n/2 + h/16 - 1. As before, let I : V -t W denote the embedding of the complete ternary tree T into G. We apply Lemma 29 to the cut defined by A = I-I (Xl UYl ) and B = I- l(X2 U Y2). Obviously, IAI = IBI = n/2, and therefore eT(A, A) + eT(B, B) 2: h/2. Assume, that eT(A, A) 2: eT(B, B). By construction, one vertex of every edge of T mapped into A is mapped to a vertex of Yl . The maximum degree of T is 4, and so there are at most 41Yl l edges of T with one endpoint mapped onto some vertex in Yl . Since 4!Yl ! < h/4 we have arrived at a contradiction: T G.

ct

REFERENCES

[IJ N. Alon and J . Spencer , The Probabilistic Method, John Wiley & Sons, Hoboken, New Jersey (2008). [2J

B. Bollobas, Extremal Graph Theory , Academic Press , London (1978).

[3]

J. Komlos, G. N. Sarkozy and E. Szemeredi, Proof of a packing conjecture of Bollobas, Combinatorics, Probability and Computing, 4 (1995), 241-255.

[4J

J. Komlos, G. N. Sarkozy and E. Szemeredi, Spanning trees in dense graphs, Combinatorics, Probability and Computing, 10 (2001), 397-416.

Bela Csaba

Ian Levitt

Department of Mathematics Western Kentucky University

Department of Mathematics Rutgers University

e-mail:

e-mail:

bela.csaba~wku.edu

ianl~ath.rutgers.edu

Judit Nagy-Gyorgy

Endre Szemeredi

Bolyai Institute University of Szeged

Department of Computer Sciences Rutgers University and Renyi Institute Budapest and lAS, Princeton

e-mail: nagy-gyorgy~ath.u-szeged.hu

e-mail:

szemered~cs.rutgers.edu

Fete of Combinatorics, pp. 139-149.

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

BETTI NUMBERS ARE TESTABLE*

GABORELEK

We prove that the Betti numbers of simplicial complexes of bounded vertex degrees are testable in constant time.

1.

INTRODUCTION

Property testing in bounded degree graphs was introduced in the paper of Goldreich and Ron [4]. In this paper we study property testing for bounded degree simplicial complexes in higher dimensions. Let d 2:: 2 be a natural number and consider finite simplicial complexes where each vertex (zero dimensional simplex) is contained in at most d edges (I-dimensional simplex). Of course, such a complex can be at most d-dimensional. What does it mean to test the p-th Betti number of such a simplicial complex? First fix a positive real number E > O. A tester takes a simplicial complex K as an input and pick C(E) random vertices. Then it looks at the C(E)neighborhoods of the chosen vertices. Based on this information the tester gives us a guess bP(K) for the p-th Betti number lJP(K) of the simplicial complex such a way that:

Prob

(

IbP(K) -lJP(K)1 IV(K)I >

) E

< E,

where V(K) is the set of vertices in K . In other words, we can estimate the p-th Betti number very effectively with high probability knowing only "Research sponsored by aTKA Grant K 69062 and 49841.

140

G. Elek

a small (random) part of the simplicial complex. The goal of this paper is to show the existence of such a tester for any c > O. That is to prove the following theorem:

Theorem 1. Betti-numbers are testable for bounded degree simplicial complexes. For graphs the O-th Betti number is just the number of components and the first Betti number can be computed via the O-th Betti number and the Euler-characteristic, hence it is not hard to see that such tester exists. For connected surfaces one can also calculate the first Betti number using just the number of vertices, edges and triangles. However in higher dimensions there is no such formula even for triangulated manifolds. Note that this paper was not solely motivated by the paper of Goldreich and Ron, but also by the solution of the Kazhdan-Gromov Conjecture by Wolfgang Luck [5]. The workhorse lemma of our paper is basically extracted from his paper using a slightly different language. It is very important to note that our proof works only for Betti numbers of real coefficients and we do not claim anything for the Betti numbers of mod-p coefficients.

2.

THE CONVERGENCE OF SIMPLICIAL COMPLEXES

Let ~d be the set of finite simplicial complexes K of vertex degree bound d that is any O-dimensional simplex is contained in at most d l-dimensional simplices . We denote by K, the set of i-simplices in K and by GK the 1skeleton of K , that is V (G K) = K 0, E (G K) = K I. A rooted r- ball of degree bound d is a simplex L E ~d with a distinguished vertex x such that for any y E V(GL) , d(x, y) ~ r, where d(x, y) is the shortest path distance of x and y in the graph GL. We denote by zr,d the rooted isomorphism classes of rooted r-balls. If K E ~d and P E V(K) then let Gr(p) be the rooted r -ball in the J-skeleton GK and Br(p) is the set of simplices a such that all vertices of a are in Gr(p). Then Br(p) is a rooted r-ball of vertex degree bound d. For a E zr,d we denote by T(K, a) the set of vertices p such that Br(p) ~ a . We set

._ IT(K,a)1

PK (a ) .-

IKol

.

We say that {Kn}~=1 C ~d is convergent (see [1] for the graph case) if

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Betti Numbers are Testable

• IKol-+ 00 • limn -+oo P«; (a) exists for any r ~ 1 and a E It is easy to see that any sequence {Ln}~=l contains a convergent subsequence.

zr,d.

c Ed such that ILol -+ 00

Let aI, a2, a3,' . . be an enumeration of all the r-balls, r have the pseudo-metric 00

ds(K,L) :=

~

1. Then we

1

L 2i IpK(ai) - pdai)l· i=l

Clearly, if {K n } ~=l be an increasing sequence of simplicial complexes they are Cauchy if and only if they are convergent. By an oriented d-complex Q we mean an element of Ed equipped with a fixed orientation for each of its simplex. Note that we do not assume that the orientations are compatible in any sense. We denote by d the set of all finite oriented d-complexes. We also define oriented r-balls, the set zr,d of all oriented r-ball isomorphism classes and the probabilities pQ(f3) accordingly. Naturally we can define the convergence of oriented d-complexes as well.

t

Proposition 2.1. Let {K n } ~=l C Ed be a convergent sequence of dcomplexes. Then one has an oriented copy Qn for each K" such that {Qn }~=l C d is convergent as well.

t

Proof. Consider LLd. random variables O(x) distributed uniformly on [0,1]. Let (ao, al, . . . , ai) E Ki be a positive orientation if

t

Then the resulting sequence {Qn} ~=l C d is convergent with probability 1. For details of the simple argument see ([3], Proposition 2.2). • Finally, we need a technical definition that we use in the subsequent sections. Let us consider the set Qi of i-simplices in an oriented d-complex Q. If a, T E Qi we say that a and T are adjacent if they have at least one joint vertex . This way we can define the shortest path distance di(u, T) (where di(u, T) = 00 is possible). The ball B:(u) is the set of i-simplices T such Ard that d; (u, T) ::; r. As above, we can define the classes Zi' and the sampling probabilities PQi (f3). Note that if a, f3 E ZT,d we say that a is isomorphic to j3 if they are isomorphic as simplicial complexes not only as metric spaces.

142

G. Elek

That is PQi ((3) defined the following way. Let (3 E ZT,d. Denote by T(Qi, (3) the number ofi-simplices T such that the simplicial complex B~(J') is isomorphic to (3, where the isomorphism preserves the root-simplex. Then

Ad

Ar d

Clearly, if {Qn}~= l C E is a convergent sequence then for any (3 E Zi' , limn -+ oo PQi ((3) exists .

3.

BETTI NUMBERS AND COMBINATORIAL LAPLACIANS

t

Let Q E d be an oriented simplicial complex . Let Oi(Q) denote the euclidean space of real functions on the l-simplices of Q. Let us consider the cochain-complex

OO(Q) ~ Ol(Q) ~ . . . Recall that if

f

E

oq(Q) then

df(ao, al , ' " ,aq+l) = f(aI, a2 ,· . . , aq) - f(ao , a2 , · .. , aq+l) + '"

+ (_l)q+l f(ao, al , . .. , aq). Then bi(Q) = dimKer di - dim 1m di-l are the Betti numbers of Q. Note that they do not depend on the choice of the orientation of Q only the underlying simplicial complex . The combinatorial Laplacians (see e.g. [2]) ~~ : Oi(Q) -t Oi(Q) are defined as D.~ := di-ld;_l + d;di . The operators ~~ are positive and self-adjoint. Also, dimKer D.~ = bi(Q) . Let us remark that by Lemma 2.5 of [2] we have the following information on the combinatorial Laplacians:

• ~~((J, T)

=f=.

0 only if (J = Tor

• ~~((J, T) is always an integer.

(J

and T are adj acent.

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Betti Numbers are Testable

4. WEAK CONVERGENCE OF PROBABILITY MEASURES First recall the notion of weak convergence of probability measures. Let {J.Ln} ~=1 be probability measures on the interval [0, K]. Then {J.Ln} ~=1 weakly converges to J.L if for any continuous function f E 0[0, K]

For an example , let J.Ln(*) = 1, then the measures J.Ln converge to the measure concentrated at the zero. Note that in this case limn -+oo J.Ln(O) =1= J.L(O) . Now let

The following theorem can be extracted from [5], nevertheless we provide a proof using only the language of real analysis , avoiding any reference to operators.

Theorem 2. Suppose that {J.Ln} ~=1 weakly converges to J.L and for any n;::: 1, c(J.Ln) ;::: O. Then limn -+oo J.Ln(O) = J.L(O). Proof. First we need some notations. For a monotone function I,

For the measures J.Ln let ern be their distribution function that is

Also, let t(.-\)

= J.L( [O,.-\n . Note that er;i = ern . Let 0'(.-\) := limsupern ( .-\) n-+oo

and

Q:(.-\)

:= lim inf ern (A) . n-+oo

The following lemma trivially follows from the definitions .

144

G. Elek

Lemma 4.1. Let j be a continuous function such that

X[O ,Aj(X) ~ j(x) ~ X[O,A+iJ(x) for any 0 ~ x

~

1

+k

K, then

Proposition 4.1. <1(A)

~

t(A) = Q:+(A) = <1+(A) .

Proof. By this lemma,

~ t(A) ~ <1 (A + ~) + ~ .

<1(A)

Hence <1(A) ~ t(A) = Q:+(A) ~ <1+(A) . Since t(A) is monotone, we have that <1(A+c) ~ t(A+c) and <1+ (A) ~ t+(A) = t(A). Thus our proposition follows.

•

The following elementary analysis lemma is proved in [6]. Lemma 4.2. Let j be a continuously differentiable function on the positive reels and Jl be a probability measure on the [0, K] interval. Suppose that F is the distribution function of u, that is Jl[O , A] = F(A) . Then for any 0< e ~ K :

l

K

-l

K

j(A) dJl =

j'(A)F(A) dA + j(K)F(K) - j(c)F(c).

Assume that K ? 1, then by the previous lemma

Since

145

Betti Numbers are Testable

we have that

That is

(1) Observe that

l

K

Q:(-\) - 0'(0) -\

<

d-\ _

e

l

K

lim infn-+oo (O"n(-\) - O"n(O)) -\ d-\.

e

By Fatou's Lemma,

(2)

l

c:

K

l

lim inf n-+oo (:n(-\) - O"n(O)) d-\:::; liminf n-+oo c:

K

O"n(-\)

~ O"n(O) d-\.

Since the right hand side of (2) is less than log K, we obtain the following inequality:

l

K

Q:(-\) ~ t(O)

o. : :; log K.

Therefore, lim>.-+o Q:(-\) = 0'(0). That is by Proposition 4.1 (3)

0'(0) = t(O) .

Since one can apply (3) for any subsequence of {O"n} ~=1 we obtain that lim O"n(O) n-+oo

= t(O).

•

146 5.

G. Elek

SPECTRAL CONVERGENCE

The goal of this section is to prove the main technical proposition of our paper. Note this is based again on the ideas in [5]. Let P : jRn --* jRn be a positive, self-adjoint operator and J1p be its normalized spectral measure that is J1p(A) := the multiplicity of A as an eigenvalue. n Note that if

IIPII :s; K

then J1p is concentrated on the interval [0, K].

t

Proposition 5.1. Let {Qn}~=l c d be a convergent sequence of oriented simplicial complexes. Then there exists K > 0 such that a) For any i ~ 1 and n ~ 1, 11~~n II :s; K . b) The normalized spectral measures of {~~n} ~=1' {J1~}~=1 weakly converge.

c) c(J1~) ~ 0 for any i, n ~ 1. Proof. To show (a) it is enough to prove the following lemma. Lemma 5.1. Let L, M > 0 be positive integers. Then if A is a n x n-matrix

of real coefficients (that is a linear operator on

jRn)

such that

• each row and column of A contains at most L non-zero elements • for each entry A ,j, then

IAi,j I ::; M

IIAII :s; 2LM.

Proof. For unit-vectors i, 9 E

I(A(f) ,g)

I =1 L

jRn

A,jf(i)9(j)I:S;M

l~i ,j~n

That is I(A(j) ,g)l~

L

If(i)g(j)I ·

l~ i,j~n

I:

(J(i)2+ g(j)2).

l~i ,j~n

Note that the number of occurrences of each f(i)2 or g(j)2 is at most L hence I(A(f) ,g) 2LM. •

I::;

147

Betti Numbers are Testable

Now let us turn to part (b) . The convergence of {J.l~}:'=1 means that

Jor P(t) dJ.l~(t) K

lim

n-+oo

exists for any real polynomial P. That is one needs to prove that

exists where {At} denotes the spectrum of the i-th Laplacian of Qn . Hence it is enough to prove that the limit of normalized traces

(4) exists. The value of bobn(o, 0-) depends only on the r-neighboorhood of 0 , therefore the convergence of the complexes {Qn} ~=1 immediately implies the existence of the limit in (4). Part (c) follows from the simple fact: If Q is a symmetric integer matrix then the product of its non-negative eigenvalues is an integer as well. Indeed, let AI, A2, . .. , Aq the list of the non-zero eigenvalues of Q with multiplicities. Let p(t) = det (tI - Q) be the characteristic polynomial of Q. Then p(t) = tSq(t), where q(O) =I O. Obviously, q is an integer polynomial, A and Iq(O)1 =

ITI?=l ql· •

6.

THE PROOF OF THEOREM

1

We need to prove the following lemma.

Lemma 6.1. Let {Kn}~=l

exists for any i

~

1.

c r;d

be simplicial complexes, then

148

G. Elek

Indeed , by Lemma 6.1 we can immediately see that for any c exists <5 > 0 such that if ds(K, L) < <5 for some K, L E ~d, then

> 0 there

(5) for any i 2: 1. By the definition of the metric d; there exists some r 2: 1 and p > 0 such that if IpM(a) - PNa)1 ~ p for any r' ~ r and a E zr' ,d then ds(M, N) < <5 for any M , N E ~d . Now let L 1 , L 2 , ••• ,Lm be a finite set of simplicial complexes such that for any finite simplicial complex K there exists 1 ~ j ~ m such that

(6) for any r' ~ r and a E zr' ,d, where r, p are the constants above. The existence of such finite system is clear from compactness. By the classical Chernoff's inequality there exists N, > 0 such that the following holds: Let M E ~d be an arbitrary simplicial complex. Pick N, random vertices of M and for any r' ~ r and a E zr',d let Q(M, a) be the number of picked vertices x such that Br,(x) ~ a . Then (7)

Prob

{I Q(~: a) -

I

PM(a) >

~ for at least one

a} <

c

Thus we have the following testing algorithm. Take the simplicial complex M as an input. Pick N, random vertices and calculate Q(M, a) for all r' ~ r, a E zr,d, where r is the constant above. Check the list £1, £2, . .. , Lm. By (7), with probability more than (I-c) we find an [) such that Q(M, a) PLj(a)1 < ~ for any a. Let bi(Lj ) be our guess. Then by (5) with probability more than 1 - c

I

So, let us prove Lemma 6.1. By Proposition 5.1 .

dim Ker ~

kn

11m -----,------,,....--'-''-n-+oo

IKil

exists . Also, by the definition of the convergence of simplicial complexes limn -+oo IJ~l)1 exists , hence the lemma, and thus our Theorem follows. •

Betti Numbers are Testable

149

REFERENCES [lJ I. Benjamini and O. Schramm , Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., 6 (2001), no. 23, 13 pp. (electronic). [2J

G. Elek, Combinatorial Heat Kernels and Index Theorems , Journal of Functional Analysis, 129 (1995) no. 1, 64-79.

[3J

G. Elek, A Regularity Lemma for Bounded Degree Graphs and Its Applications: Parameter Testing and Infinite Volume Limits (preprint), URL: http://arxiv.org/pdf/0711.2800v2.

[4J

O. Goldreich and D. Ron, Property Testing in Bounded Degree Graphs, Algorithmica, 32 (2002), 302-343 .

[5J W. Liick, Approximating L 2-invariants by their finite-dimensional analogues, Geom. Funct. Anal. , 4 (1994), no. 4, 455-481. [6J W. Liick, L 2-torsion and 3-manifolds, Low-dimensional topology (Knoxville, TN, 1992), 75-107. Conf. Proc . Lecture Notes Geom. Topology, III , Internat. Press (Cambridge, MA, 1994).

Gabor Elek Alfred Renyi Institute H-1053 Redltcnoda u. 13-15. Budapest Hungary a-mail :

elek~renyi .hu

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 151-172.

RIGID AND GLOBALLY RIGID GRAPHS WITH PINNED VERTICES

TIBOR JORDAN*

We consider rigid and globally rigid bar-and-joint frameworks (resp. graphs) in which some joints (resp . vertices) are pinned down and hence their positions are fixed. We give an overview of some old and new results of this branch of combinatorial rigidity with an emphasis on the related optimization problems. In one of these problems the goal is to find a set P of vertices of minimum total cost for which the positions of all vertices become uniquely determined when P is pinned down. For this problem, which is motivated by the localization problem in wireless sensor networks , we give a constant factor approximation algorithm.

1. INTRODUCTION A bar-and-joint framework (or simply framework) (G, p) in d-space is a graph G = (V, E) and a map p : V -+ ]Rd. We also say that (G,p) is a ddimensional realization of G. We can think of the edges and vertices of Gin the framework as rigid (fixed length) bars and universal joints, respectively. An infinitesimal motion is a map x : V -+ ]Rd satisfying

for all edges ViVj E E . The initial velocities obtained by differentiating a smooth motion of the (vertices of the) framework which preserves the edge lengths give rise to an infinitesimal motion of (G,p). The rigidity matrix "This work was supported by the Mobile Innovation Centre, funded by the Hungarian National Office for Research and Technology.

152

T. Jordan

of the framework (G,p) is the matrix R(G,p) of size lEI x dlVl, where, for each edge e = ViVj E E, in the row corresponding to e, the entries in the two columns corresponding to vertices i and j contain the d coordinates of (p( Vi) - p( Vj)) and (p( Vj) - p( Vi)) , respectively, and the remaining entries are zeros. Example. The rigidity matrix of the framework of Figure 1 is as follows. The rows correspond to edges ab, be, ca, cd, in this order, and consecutive pairs of columns correspond to vertices a, b, c, d.

Thus x (viewed as a vector in ]RdlVl) is an infinitesimal motion if and only if R(G,p)x = O. Each translation and rotation of ]Rd gives rise to a smooth motion of (G, p) and hence to an infinitesimal motion of (G, p). These rigid motions of]Rd give rise to a subspace of dimension (d!l) in the null-space of R( G, p). Hence Lemma 1.1 [31, Lemma 11.1.3J. Let (G,p) be a framework in

(1)

]Rd. Then

rankR(G,p)::=; S(n,d),

I

where n = V(G)I and nd- ( d +2

1)

if

n~d+2

S(n,d) =

n) (2

if n::=;d+l.

We say that a framework (G, p) is infinitesimally rigid in ]Rd if the rank of its rigidity matrix R(G,p) is maximum, i.e. if equality holds in (1). A framework is rigid if it has no non-trivial smooth motions. Thus infinitesimal rigidity is a sufficient condition for rigidity. It is known that for "generic" frameworks the two notions are the same. We refer the reader to [10,31,32] for more details on the theory of rigid frameworks.

153

Rigid and Globally Rigid Graphs with Pinned Vertices

b d

a

(a)

d

d

(b)

(c)

Fig. 1. A framework in ~2 on four vertices (left) . The coordinates of the vertices are as

follows: p(a) = (0,0) , p(b) = (0,1), p(c) = (1,1), p(d) = (2,0) . Since rankR(G,p) = 4, to fix the framework one needs tracks of co-dimension four in total, which can be achieved by two one-dimensional tracks and a pin (middle) or two pins (right) .

2\VI -

2.

RIGID FRAMEWORKS WITH PINNED VERTICES

Let G = (V, E) be a graph and consider a d-dimensional realization (G, p) of G. We may fix (G,p) in jRd by restricting the infinitesimal motions of its vertices to given subspaces of jRd. Suppose that for all vertices v E V we are given a subspace U(v) S;;; jRd , generated by a subset of the standard basis of jRd . We call U (v) the track of v and we say that (G, p) is fix ed by the given set of tracks if the only infinitesimal motion x of (G,p) satisfying x( v ) E U(v) for all v E V is the zero vector x = O. In most cases we shall be interested in the special case when each track is either zero- or ddimensional. We say that P S;;; V is a pinning set if (G ,p) is fixed by the tracks U(v) = {O} if v E P , U(v) = jRd if v rt. P. We also say that the vertices in P are pinned down, or that each vertex of P is a pin. The following lemma establishes the connection between tracks (pins) that fix a framework and its rigidity matrix (see also [26, Statement 8.2.1]). Note that each track U(v) of dimension k, 0 ::; k ::; d, corresponds naturally to a subset of size k of the d columns of the rigidity matrix which belong to v. Lemma 2.1. Let (G,p) be a framework in jRd , let U = (U(v) : v E V) be a family of tracks, and let Ru be the matrix consisting of all columns of R( G, p) which correspond to the tracks U(v) , v E V. Then

(i) U fixes (G ,p) if and only if the columns of Ru are linearly independent, (ii) P is a pinning set if and only if the dlV -PI columns of R(G,p) indexed by V - P are linearly independent.

154

T. Jordan

One may ask for an optimal family of tracks that fixes a given framework by using the least possible total restriction, i.e. an assignment U = (U (v), v E V) for which U fixes (G,p) and

L (d - dim U(v)) vEV

is mmimum. By Lemma 2.1(i) an optimal family of tracks is easy to find by using a greedy algorithm to identify a maximum size independent set of columns in R(G,p). Furthermore, the optimum is unchanged if we restrict the matrix to a maximum size set of independent rows (or if we consider the corresponding subgraph of G). It is also clear that min {

L

(d - dimU(v)) : U fixes (G,p)} =

dlVl- rankR(G,p).

vEV

We obtain a much more difficult problem if we impose restrictions on the dimension of the tracks. This is the case, for example, when we consider pinning sets. The pinning number, pind(G,p), of (G,p) is defined to be the size of a smallest pinning set for (G,p) . For d = 2 Lemma 2.1(ii) implies that the smallest pinning set problem can be formulated as a matroid matching problem in a linearly represented matroid and hence pin, ( G, p) can be computed in polynomial time by using the algorithm of Lovasz [21]. A combinatorial formula for pin2(G,p) was also given by Lovasz [22]. Mansfield [25] proved that the problem of computing pin, (G, p) for a framework (G,p) is NP-hard. On the other hand, a recent result of Szabo [30] shows that there exist tractable cases even in dimensions larger than two. Theorem 2.2 [30]. The following problem is polynomial time solvable. Given a framework (G,p) in ~3 and a partition V = VI U V2 , find a family U = (U (v) : v E V) of tracks minimizing 2:vEV ( d - dim U (v)) such that U fixes (G,p) and (i) dimU(v) E {O, 1,3} for v E VI, and (ii) dim U(v) E {O, 2, 3} for v E V2 . The algorithm in [30] also uses Lovasz' matroid matching algorithm as a subroutine. If we are also given a cost function on the vertices, we may look for a pinning set of minimum total cost. Baudis et al [2] proposed an approximation

Rigid and Globally Rigid Graphs with Pinned Vertices

155

algorithm for this more general problem. For d-dimensional frameworks the approximation guarantee is 1 + ~ + + ... + ~ . Their algorithm is based on a general result about minimum cost spanning sets in d-polymatroids, see also Section 8.

!

3. RIGID GRAPHS WITH PINNED VERTICES The rigidity matrix of a d-dimensional framework (G, p) defines the rigidity matroid of (G,p) on the ground set E where a set of edges F ~ E is independent if and only if the rows of the rigidity matrix indexed by Fare linearly independent. A framework (G, p) is generic if the set of coordinates of the points p(v), v E V , is algebraically independent over the rationals. Thus, since the entries of the rigidity matrix are polynomial functions with integer coefficients, any two generic d-dimensional frameworks (G, p) and (G, q) have the same rigidity matroid. We call this the d-dimensional rigidity matroid 1?d(G) of the graph G. We denote the rank of 1?d(G) by rd(G). We say that a graph G = (V, E) is generically infinitesimally rigid, or simply rigid, in ]Rd if rd(G) = S(n, d). We say that a graph G = (V, E) is independent in ]Rd if E is independent in 1?d(G). It is not difficult to see that 1?1 (G) is the cycle matroid of G. It remains an open problem to find good characterizations for independence or, more generally, the rank function in the d-dimensional rigidity matroid of a graph when d 2': 3. Similarly, any two generic d-dimensional frameworks on G have the same pinning number. Thus we may define the pinning number of G, pind(G), as the pinning number of (G,p) of any generic framework (G,p) in ]Rd. It is easy to see that pind(G) :::; pind(G,p) for all frameworks (G,p) . The next lemma implies that computing the pinning number of G is the same as finding a smallest complete graph whose addition to G makes it rigid . For a set P ~ V (G) let G + K (P) denote the graph obtained from G by joining all pairs of non-adjacent vertices of P .

Lemma 3.1. Let G = (V, E) be a graph and P ~ V with !PI 2': d. Let (G, p) be a generic realization of G in ]Rd . Then P is a pinning set for (G, p) if and only if G + K(P) is rigid in ]Rd. Proof. Let G' = G + K(P). First suppose that G' is rigid and consider the rigidity matrix R(G',p). Since G' is rigid, the only solutions u to the equation R(G', p)u = 0 are from rigid congruences of IRd • Thus, since (G', p)

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is generic, each non-zero solution leaves at most (d - 1) vertices fixed i.e, has at most (d -1) zero entries. Suppose R (G[V - P], p) has linearly dependent columns. Then we can find a non-zero solution u' to R( G[V - P], p) u' = O. By extending u' to u by putting 0 in the components corresponding to P we obtain a non-zero solution to R(G', p)u = 0 with at least IPI ~ d zeros, a contradiction. Thus P is a pinning set by Lemma 2.1(ii) . Now suppose that P is a pinning set and order the columns of R =

R(G',p) so that the columns of P come first and the rows of E" = E(G'[p]) come first. (Then the upper right quarter is 0.) Hence r(R) ~ r( R[P,E"]) + P, E - E"]) = dlPI - (d!l) + dlV - PI = dlVl - (d!l) (by using Lemma 2.1(ii) and that G'[P] is rigid and IPI ~ d). Thus G' is rigid. •

r( R[V -

Next we show that in the pinning problem we may assume that G is independent.

Lemma 3.2. Let F ~ E be a maximal edge set of G = (V, E) for which H = (Y; F) is independent in Rd. Then (i) each pinning set of G is a pinning set of H,

(ii) pind(H) = pind(G). Proof. To prove (i) suppose, for a contradiction, that there exists a pinning set P of G for which H + K{P} is not rigid. Since G + K{P} is rigid , we have rd( G + K(P)) > Td(H + K(P)) , which implies that there is an edge e E E+E( K(P)) -(F+E( K(P))) = E-F for which F+e is independent, contradicting the maximality of F . This proves (i), from which (ii) follows immediately. • It follows from the observations above that the pinning problem in graphs (or in generic frameworks) can be attacked by purely combinatorial methods provided good characterizations for independent and rigid graphs are available. This is the case when d = 2 and we shall discuss this approach in the 2-dimensional case in the forthcoming sections.

Mansfield [25] proved that the problem of computing pin3 (G) for a graph G is NP-hard (see also [7] for a different proof), so the pinning problem in higher dimensions seems untractable. The following related result, however, might be useful in a different context as it points to a connection between high connectivity and rigidity in 3-space. It may be considered as a first step towards the Lovasz-Yemini conjecture [23], which asserts that sufficiently highly connected graphs are rigid in 3-space (and hence their pinning number is three) .

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Theorem 3.3 [14].

Let G = (V, E) be a 10-connected graph.

Then

pin3(G) ~ 3'r' + 4.

4. THE TWO-DIMENSIONAL RIGIDITY MATROID In the rest of the paper we will be concerned with the case when d = 2 and suppress the subscript d accordingly. In this section we first describe the characterization of independent and rigid graphs and prove some additional structural results which may also be useful in the solution of the pinning problem. For X ~ V let Ec(X) denote the set, and ic(X) denote the number of edges in G[X], that is, in the subgraph induced by X in G. We say that a graph G is sparse if ic(X) ~ 21Xj - 3 for all X ~ V with IXI ~ 2. It is easy to show, by using the 2-dimensional case of Lemma 1.1, that independent graphs are sparse . Laman [18] proved that this necessary condition is also sufficient. Theorem 4.1 [18]. A graph G = (V, E) is independent if and only if G is sparse.

A cover of G = (V, E) is a collection X = {Xl , X 2 , • • • , X t } of subsets of V, each of size at least two, such that UiE(X) = E. The cover is said to be thin if \X i n Xj I ~ 1 for all i =1= j. The value val (X) of the cover is E~=l (2I X il - 3). Let X be a thin cover of G and let F ~ E be a set of edges for which H = (V,F) is sparse. Then we have IF n EC(Xi) ~ 21Xi l - 3 for all 1 ~ i :s; t. Thus

I

(2)

IFI :s; val (X) .

We define a rigid component of a graph G = (V, E) to be a maximal rigid subgraph of G. By the "plane gluing lemma" (see [31, Lemma 3.1.4]), which says that the union of two rigid graphs with at least two vertices in common is rigid, it follows that the vertex sets of the rigid components form a thin cover of G. In a sparse graph H we call a set X ~ V (H) critical if iH(X) = 2\X I - 3 holds. It follows from the gluing lemma that if X ,Y c V(H) are critical sets in H with IX n YI ~ 2 then Xu Y is also critical (see also [13, Le mma 2.3]) .

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Lemma 4.2. Let G = (V; E) be a graph, let F ~ E be a maximal edge set in G for which H = (V, F) is sparse. Then the family X = {X I , X 2, ... , X t } of maximal critical sets in H satisfies that (a) X is a thin cover of G with IFI = val (X), (b) X is equal to the family of vertex sets of the rigid components of G. Proof. (a) The maximality of the critical sets implies that IXi n Xjl :::; 1 for all 1 :::; i < j :::; t. Since every single edge of F induces a critical set , it follows that X = {Xl, X 2 , •• • , Xt} is a thin cover of H . Thus t

IFI =

t

L IEH(X I = L (2I Xil- 3). i)

I

I

To complete the proof we show that X is a cover of G as well. Choose uv E E - F. Since F is a maximal sparse subset of E , F + uv is not sparse. Thus there exists a set X ~ V such that u , v E X and iH(X) = 21XI - 3. Hence X is a critical set in H. This implies that X ~ Xi and hence uv E Ec(Xi ) for some 1 :::; i :::; t. (b) Clearly, G[Xi ] is rigid for all 1 :::; i :::; t by Theorem 4.1. Suppose that H[G] is not critical, where C is the set of vertices of some rigid component of G. Thus PI :::; 21GI - 4, where J = E( H[G]). Since G[Xi ] is rigid, it follows from the gluing lemma that X' = {Xi EX : IXi n CI ~ 2} is a thin cover of arC] with IJI = val (X') . Thus we can use (2) to deduce that for any subset F' ~ E( G[C]) which induces a sparse subgraph on vertex set C we have IF'I :::; val (X') = IJI :::; 21CI - 4, contradicting the fact that G[G] is rigid. This completes the proof of (b). • Lemma 4.2(a) shows that for the edge set of a maximal sparse subgraph and for its maximal critical sets we have equality in (2). This implies the following rank formula of the rigidity matroid, due to Lovasz and Yemini, and shows that a maximum size sparse edge set can be found greedily. Theorem 4.3 [23]. Let G = (V, E) be a graph. Then r(G) = min { val (X) : X is a thin cover of G} . A maximum size sparse edge set (and the rigid components of a graph) can be found in time O(n 2 ) , see e.g. [4]. We may simplify the min-max formula of Theorem 4.3 when the graph is obtained from a sparse graph by 'pinning' a set of vertices. For a set X ~ V let e(X) denote the number of edges with at least one end-vertex in X .

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Lemma 4.4. Suppose that G = (V, E) is a sparse graph and let P with IPI 2: 2. Let G' = G + K(P). Then

~

V

r(G') = min 21ZI- 3 + e(V - Z). pc;,z

Proof. Let Z ~ V with P ~ Z and consider the thin cover Z = {Z

{{u,v} : uv

E

E - E(Z)}} of G'. Then r(G') ::; val(Z) =

u 2\ZI- 3 +

e(V - Z).

To see that equality holds for some Z ~ V choose a maximal edge set F in G' for which H = (V, F) is sparse and P is a critical set in H. Such an F can be constructed by extending the edge set F' of a minimally rigid subgraph of the complete graph G'[P]. Let X be the family of maximal critical sets of H. By Lemma 4.2(a) and Theorem 4.3 we have r(G') = val (X). Since P is critical in H, there is a set Z E X with P ~ Z. Thus, since G is sparse and all edges of K(P) are covered by Z, we have ic(X) = iH(X) = 21XI - 3 for all X E X - Z. Hence r(G') = val (X) = 2IZI-3+ L:XEX-Z ic(X) = 2IZI-3+e(V -Z) , which completes the proof.

•

5.

OPTIMAL FAMILIES OF TRACKS AND SMALLEST PINNING SETS

Let G = (V, E) be a graph. First consider the problem of finding an optimal family of tracks, U = (U(v) : v E V) , which fixes (G, p) for a generic realization of G in ~2 . As we have observed earlier, we may assume that G is independent (or equivalently, that G is sparse). Thus lEI = 21V1 - k for some integer k ~ 3. It is also clear that L:VEV (2 - dim U (v)) = k for an optimal family of tracks. The following algorithm, due to Lee et al. [19], determines an optimal family of tracks in O(n 2 ) time. It uses k - 2 onedimensional tracks (also called sliders) and one pin to fix (G, p). (For the remaining vertices the tracks are two-dimensional.) The algorithm works as follows. First identify the rigid components of G. Mark one of the components, say 0 , as the base. For some edge uv in C assign a pin to u and a slider to v. This fixes the base. Then repeat the following until one rigid component remains: pick an edge ij which leaves the base and assign a slider to j. Update G by adding a new edge jk , where

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ik is an edge in the base. Replace the base 0 by the rigid component of the updated graph containing uv . The correctness of this algorithm follows from the fact that if 0' is a rigid component that shares vertex i with 0 then the only motion of 0' with respect to 0 is a rotation about i. Since the framework is generic, assigning a slider to j eliminates this motion and hence the distance between j and k becomes fixed. Thus every iteration increases the rank by one and therefore the algorithm will terminate with a rigid graph after adding at most k - 3 sliders (not counting the pin and the slider added to fix the original base). The algorithm, when applied to (a generic realization of) the graph of Figure l(a) , may give the family of tracks shown by Figure l(b). We remark that combinatorial characterizations for the generic rigidity of bar-and-slider frameworks (which are bar-and-joint frameworks equipped with sliders at given joints) have been given in [19], and also in [17], where the authors consider the version in which the directions of the slider lines are also given. Next consider the pinning problem. Lemma 5.1. [7] Let G = (V, E) be a sparse graph and let P ~ V with IPI ~ 2. Then P is a pinning set for G if and only if 21XI ::; e(X) for all X s:; V - P . Proof. Suppose, for a contradiction, that P is a pinning set and 2!XI > e(X) for some X ~ V - P and let Z = V - X . Then X = {Z u {{u, v} : uv E E-E(Z)}} is a thin cover ofG+K(P) with val (X) ::; 2IZI-3+e(X) < 21V1 - 3. Thus, by Theorem 4.3, G + K(P) is not rigid. Hence P is not a pinning set by Lemma 3.1, a contradiction. Now suppose 21XI ::; e(X) for all X ~ V - P. It follows from Lemma 4.4 that there is a thin cover X of G + K(P) with P ~ Z for some Z E X and r( G + K(P)) = val (X) = 21ZI- 3 + e(V - Z) . Since e(V - Z) ~ 21V - ZI this gives val (X) = 21V1- 3. Hence G + K(P) is rigid and, by Lemma 3.1, P is a pinning set. _ Thus finding a smallest pinning set is equivalent to finding a largest set V for which e(X) ~ 21XI for all X ~ Y. This can be formulated as a matching problem in an auxiliary graph (see Fekete [7]) and can be solved in O(n 2 ) time . Fekete [71 also provides a min-max formula for pin2(G). Makai and Szabo [24] deduce this formula by using polymatroidal methods.

Y

s:;

Rigid and Globally Rigid Graphs with Pinned Vertices

161

We note that Servatius, Shai, and Whiteley [28] consider a different version of the pinning problem and provide a characterization and a decomposition result for the so-called pinned isostatic graphs .

6.

THE NETWORK LOCALIZATION PROBLEM

In the network localization problem the locations of some nodes (called anchors) of a network as well as the distances between some pairs of nodes are known, and the goal is to determine the location of all nodes. This is one of the fundamental algorithmic problems in the theory of wireless sensor networks and has been the focus of a number of recent research articles and survey papers, see for example [1] .

Fig . 2. The distance graph and the grounded graph of a network on six nodes, including

four anchor nodes. The anchor nodes are in boxes. The network is uniquely localizable since it has at least three anchors and its grounded graph is globally rigid . This is a smallest anchor set which can guarantee unique localizability for the given set of distances.

A natural additional question is whether a solution to the localization problem is unique. The network, with the given locations and distances, is said to be uniquely localizable if there is a unique set of locations consistent with the given data. As we shall see, the unique localizability of a two-dimensional network, whose nodes are in generic position, can be characterized by using results from graph rigidity theory. In this case unique localizability depends only on the combinatorial properties of the network: it is determined completely by the distance graph of the network and the set of anchors, or equivalently, by the grounded graph of the network and the number of anchors. The vertices of the distance and grounded graph correspond to the nodes of the network. In both graphs two vertices are connected by an edge if the corresponding distance is explicitly known. In the grounded graph we have additional edges: all pairs of vertices corresponding

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to anchor nodes are adjacent. See Figure 2. The grounded graph represents all known distances, since the distance between two anchors is determined by their locations . Before stating the basic observation about unique localizability we need some additional terminology. It is convenient to investigate localization problems with distance information by using frameworks, the central objects of rigidity theory. Two frameworks (G , p) and (G, q) are equivalent if corresponding edges have the same lengths, that is, if II p(u) - p(v) II = II q(u) - q(v) I holds for all pairs u, v with uv E E, where 11.11 denotes the Euclidean norm in ~d. Frameworks (G,p), (G,q) are congruent if IIp(u) - p(v) II = IIq(u) - q(v) II holds for all pairs u , v with u , v E V . This is the same as saying that (G, q) can be obtained from (G,p) by an isometry of ~d. We shall say that (G,p) is globally rigid, or that (G , p) is a unique realization of G in ~d , if every framework which is equivalent to (G ,p) is congruent to (G ,p) . We say that a graph G is globally rigid in ~d if every (or equivalently, if some) generic realization of G in ~d is globally rigid. The next observation shows that unique localizability and global rigidity are, in some sense, the same. Lemma 6.1 [1 , 29] . Let N be a network in ~d consisting of m anchors located at positions PI , . . . ,Pm and n - m ordinary nodes located at Pm+1 ,' " ,Pn' Suppose that there are at least d + 1 anchors in general position. Let G be the grounded graph of N and let P = (PI, .. . ,Pn). Then the network is uniquely localizable if and only if (G,p) is globally rigid. Globally rigid graphs in ]R2 have been characterized by Jackson and Jordan [13], relying on earlier results of Hendrickson [11] and Connelly [6] . We say that a graph G is redundantly rigid in ]R2 if G - e is rigid in ]R2 for all e E E(G). Theorem 6.2 [13]. Let (G ,p) be a generic framework in ~2 . Then (G ,p) is globally rigid if and only if G is a complete graph on at most three vertices or G is 3-connected and redundantly rigid. Theorem 6.2 implies that global rigidity is indeed a generic property. It also implies that global rigidity can be tested in O( n 2 ) time. We shall consider the minimum cost anchor set problem in which the goal is, given the set of known distances in a network and a cost function on the nodes, to designate a minimum cost set of anchor nodes which makes the network uniquely localizable. Lemma 6.1 and Theorem 6.2 imply that

Rigid and Globally Rigid Graphs with Pinned Vertices

163

for generic networks we may reformulate the above problem in the following purely combinatorial form: Given a graph G = (V, E) and a function c : V -+ ~+, find a set P ~ V, !PI ~ 3, for which G + K(P) is 3-connected and redundantly rigid, and c(P) = LVEP c(v) is minimum. In the next sections first we shall show that a relaxed version (in which the requirement is that the rigidity matroid of G+K(P) must be connected) can be formulated as a matroid optimization problem. Then, based on this formulation, we shall develop a polynomial time approximation algorithm for the minimum cost anchor set problem. Note that the complexity status of each of the above problems is still open.

7.

GRAPHS WITH A CONNECTED RIGIDITY MATROID

Given a matroid M = (E ,I), we define a relation on E by saying that e, fEE are related if e = f or if there is a circuit C in M with e, fEe. It is well-known that this is an equivalence relation. The equivalence classes are called the components of M. If M has at least two elements and only one component then M is said to be connected. We say that a graph G = (V, E) is M -connected if R(G) is connected. For example, K3,m is M-connected for all m ~ 4. The M -components of G are the subgraphs of G induced by the components of R(G). It is easy to see that the M-components are pairwise edge-disjoint induced subgraphs. Theorem 6.2 and the following result show that M-connectivity is in between redundant rigidity and global rigidity. Theorem 7.1 [13]. Let G be a graph. Then

(a) if G is M -connected then G is redundantly rigid, and (b) if G is 3-connected and redundantly rigid then G is M -connected. Since the M-components of G are redundantly rigid by Theorem 7.1, the partition of E(G) given by the M-components is a refinement of the partition given by the rigid components, see Figure 3. The rigidity matroid of a graph G is the direct sum of the rigidity matroids of either the rigid components of G or the M-components of G. Furthermore, the vertex sets of the components in each of the above decompositions form a thin cover

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x «'---«

Fig. 3. This graph is rigid so has exactly one rigid component. It has five M -connected components: each of the three copies of K 4 , and the remaining two copies of K2.

of G with minimum value. This minimum value is equal to the rank of R(G) by Theorem 4.3. The following lemma is easy to prove by standard matroid techniques . Lemma 7.2. Let M = (E , r) be a matroid on ground set E with rank function r and let E I , E2, . .. .E; be the components of M. Then

2::i r(Ei) , and (ii) if r(E) = 2::1 r(Fi ) for some partition F I , F2, .. . , Fq of

(i) r(E) =

component of M for some 1 ::; i ::; t, then E i

~

E and E; is a Fj for some 1 ::; j ::; q.

The next lemma shows how this general result can be formulated in terms of subgraphs and covers in the special case when the matroid is the rigidity matroid of a graph. We say that a cover is non-trivial if it contains at least two sets . Lemma 7.3 [9]. G = (V, E) is M-connected ifand only if val (X) 2: for all non-trivial covers X of G.

21V1-2

Proof. First suppose that G is M-connected. Then G is rigid, and hence val (X) 2: 21V1 - 3 for all covers X of G by (the easy direction of) Theorem 4.3. Suppose that val (X) = 21V1 - 3 for some non-trivial cover X = {Xl , X2, .. . , X q } of G. Let Fi = E( G[X i ]) , 1 ::; i ::; q. We have r(Fi ) = 21Xi l - 3 for all 1 ::; i ::; q, as X is a cover of G which minimizes val (X) . Thus r(E) = val (X) = 2::1 r(Fd , which contradicts Lemma 7.2(ii). To prove the other direction suppose that val (X) 2: 21V1- 2 for all nontrivial covers X of G, but G is not M-connected. Let HI, H2, . . " H, be the

Rigid and Globally Rigid Graphs with Pinned Vertices

165

M-components of G. Lemma 7.2(i) now implies that 2JVI - 3 ~ r(E) = L~ r( E(Hi)) = L~ (2JV(Hi )l- 3) . Thus , since each edge of G belongs to some M-component and t ~ 2, X = {V(H1 ) , V(H2),"" V(Ht ) } is a nontrivial cover of G with val (X) :::; 2JVI- 3. This contradicts our assumption.

• 7.1. M-connected graphs with pinned vertices

In the M -connected pinning problem the goal is to find a (smallest) set P ~ V for which G+K(P) is M-connected. The following lemma establishes the connection between the feasible solutions of the M -connected pinning problem and the M-components of G. Lemma 7.4 [9]. Let G = (V, E) be a graph, let 1{ = {HI, H2, .. " Ht } be the M-components of G, and let P ~ V with IPI ~ 4. Then G + K(P) is M -connected if and only if

(3)

2JVI- 2:::; 21ZI- 3 +

L

(2JV(Hi )J- 3)

HiE1iz

holds for all Z c V with P V(Hi) n (V - Z) =1= 0} .

~

Z, Z

=1=

V, where Hz

Proof. First suppose that G + K (P) is M -connected. Since every edge of G belongs to an M-component of G and P ~ Z, it follows that {Z}U {V(Hi ) : Hi E 1{, V(Hi) n (V - Z) =1= 0} is a cover of G + K(P). This cover is non-trivial, since Z =1= V. Thus (3) follows from Lemma 7.3. To prove the other direction suppose , for a contradiction, that (3) holds but G' = G + K (P) is not M -connected . Let 1{' = {Hi, H2,... , H~} denote the M -components of G' . Since complete graphs on at least four vertices are M-connected, and !PI ~ 4, it follows that G'[P] is M-connected. Thus there is an M-component of G', say Hi, for which P ~ V(Hi) . Let Z' = V(Hi) and 1{z, = {Hi E 1{ : V(Hi) n (V - Z') =1= 0}. Note that Z' =1= v. Claim 7.5. Let X ~ V be a set of vertices. Then X = V(Hj) for some M -component Hj of G' with 2 :::; j :::; q if and only if X = V(H) for some HE 1{z, .

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Proof. First consider an M-component Hj E 'H' with j ~ 2 and let X = V(Hj). Since P ~ Z' and Hf is an induced subgraph of G' which has no edge in common with Hj, it follows that G[X] is M-connected and X n (V - Z') -=J 0. Thus X = V(H) for some H E 'Hz, . Next consider an M-component Hi E 'Hz' of G and put X = V(Hi). G'[X] is clearly M-connected. For a contradiction suppose that there is an M-component Hj of G' with V(Hj) = Y ~ V for which X is a proper subset of Y. Then IY n Z'I ~ IY n PI ~ 2 must hold. Since X n (V - Z') -=J 0, we have j ~ 2. This contradicts the fact that the M-components of G' are pairwise edge-disjoint. Thus G'[X] is an M-component of G', which completes the proof. • By using Claim 7.5 and Lemma 7.2(i), and by applying (3) with Z = Z', we obtain 21V1- 3 ~ r(G') = 21 V(HDI - 3 +

L

(21 V(Hi)1 - 3) ~ 21V1- 2,

HiE1l z '

a contradiction. • Let G = (V, E) be a graph and let 'H = {HI, H 2 , ... , Ht} be the Mcomponents of G. Let H(G) = (V, £) be the hypergraph which contains 21 V(Hi)1 - 3 copies of the hyperedge V(Hi) for each Hi E 'H, 1 ::; i ::; t. Note that since the M-components are rigid it follows from Lemma 7.2(i) that 1£1 = r( G) ::; 21V1- 3. By letting Y = V - Z in Lemma 7.4 and using the above definitions we obtain: Lemma 7.6. Let G = (V, E) be a graph, let'H = {HI, H 2 , •. • , Ht} be the M-components of G, and let P ~ V with IPI ~ 4. Then G + K(P) is M -connected if and only if

(4) holds for all non-empty subsets Y ~ V - P, where eH(G)(Y) denotes the number of hyperedges e E £ with en Y -=J 0. A hypergraph F = (V, F) satisfying IU F'I ~ IF'I + 1 for all 0-=J F' ~ F is called a hyperforest. Inequality (4) can be reformulated in terms of hyperforests as follows. Let L(G) = (W,U) be the hypergraph obtained from the dual hypergraph of H (G) by duplicating every hyperedge". For a lConsider the bipartite incidence graph C* of H(C) and split each vertex u E V into two vertices u',u". See Figure 4. Then we obtain the bipartite incidence graph of L(C) by interchanging the color classes.

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Fig. 4. The bipartite incidence graph of L(G) , where G is th e graph of Figure 2.

set X ~ V let U(X) denote the set of hyperedges corresponding to X in L(G) . Thus IU(X)! = 21XI.

Lemma 7.7. Let G = (V, E) be a graph and let P ~ V with P satisfies (4) if and only if U(V - P) is a hyperforest.

IPI 2: 4.

Then

Lorea [20] proved that the edge sets of the subhypergraphs of a hypergraph H' which are hyperforests form the family of independent sets of a matroid. A matroid arising this way is called the circuit matroid of the hypergraph H' and will be denoted by MH' , We call a matroid which is the circuit matroid of a hypergraph a hypergraphic matroid. Let M be a matroid on ground-set 8 and suppose that 8 is partitioned into a set A of pairs. A subset M ~ A is a matroid matching if the union of the pairs in M is independent in M . In the matroid matching problem the goal is to find a largest matroid matching, see [27, Chapter 43] . Lovasz [21] has shown that this problem may require exponential time in general but can be solved polynomially if the matroid is represented by a set of vectors in some linear space. By the above discussion and Lemma 7.7 it follows that the problem of finding a smallest set P for which G + K(P) is M-connected can be formulated as finding a largest matroid matching in the hypergraphic matroid ML(G)' in which the doubled hyperedges form the pairs . Hypergraphic matroids are known to be linear, but it is not known how to find a suitable linear representation. The complexity status of the matroid matching problem in hypergraphic matroids is still open. Nevertheless, this formulation can be used to design a randomized algorithm, see [8], or a constant factor approximation algorithm which works for the more difficult minimum cost version as well. To describe the approximation algorithm we need the following concepts. A 2-polymatroid is a pair (8, J), where 8 is a finite ground set and f is a nonnegative , monotone increasing, integer-valued , and submodular function on

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the subsets of 8, for which f(s) :::; 2 for all s E 8. A set X if f(X) = f(8) .

~

8 is spanning

Let G = (V, E) be a graph and X ~ V . Let us define b : 2v ---+ Z+ by letting

(5)

b(X) = r*(U(X)) ,

where r" is the rank function of the matroid dual of the hypergraphic matroid ML(G)' Then (V, b) is a 2-polymatroid. For a spanning set X ~ V we have r*(U(X)) = b(X) = b(V) = r" (U (V)). Thus X is spanning if and only if the set corresponding to U(V -X) is independent in ML(G)' Together with Lemmas 7.6 and 7.7 this implies:

Lemma 7.8. Let G = (V, E) be a graph and P ~ V with IPI ~ 4. Then G + K(P) is M -connected if and only if P is a spanning set of the 2polymatroid (V, b).

8.

Low COST ANCHOR SETS IN UNIQUELY LOCALIZABLE NETWORKS

Given a 2-polymatroid (8, f) and a cost function c : 8 ---+ 1R, the minimum cost spanning set problem is to find a spanning set X of the 2-polymatroid that minimizes c(X) = 'L:sEx c(s). Baudis et al. [2] verified that the GSS (Greedy Spanning Set) algorithm is a constant factor approximation algorithm for this problem. Algorithm GSS starts with X = 0 and, as long as f(X) < f(8) holds, adds a new element s to X for which f(X:S~~f(X) is minimum.

Theorem 8.1 [2]. Let (8, f) be a 2-polymatroid, let c : 8 ---+ IR be a cost function, and let X opt be a spanning set of minimum cost. Then

where X is the spanning set output by algorithm GSS. Lemma 7.8 and Theorem 8.1 give rise to a ~-approximation algorithm for the minimum cost M -connected pinning problem. To see this it remains

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to note that by using bipartite matching algorithms it is easy to test independence in ML(G) and evaluate b(X) for some X ~ V in polynomial time. To obtain an approximation algorithm for the minimum cost anchor set problem (defined in Section 6) we also need a subroutine for the minimum cost 3-connected pinning problem. Let H = (V, E) be a 2-connected graph. For some X ~ V let N(X) denote the set of neighbours of X . We say that X c V is tight if (X) = 2 and X U N (X) =!= V. The following lemma shows that a minimum cost set P' for which H + K(PI) is 3-connected can be found, in a greedy manner, in linear time.

IN I

Lemma 8.2. Let H = (V, E) be 2-connected and let P' ~ V . Then H + K(PI) is 3-connected if and only if P' n X =!= 0 for all minimal tight sets X of H. Furthermore, the minimal tight sets of H are pairwise disjoint and can be found in linear time.

Recall that redundant rigidity and M -connectivity are the same for 3connected graphs by Theorem 7.1. Thus, by combining the approximation algorithm for the minimum cost M-connected pinning problem and the algorithm for the minimum cost 3-connected pinning problem we obtain a constant factor approximation algorithm for the minimum cost anchor set problem. Theorem 8.3. There is a polynomial time ~-approximation algorithm for the minimum cost anchor set problem. Proof (sketch). Let c" denote the optimum value. By checking all feasible solutions P ~ V with IPI = 3 we may suppose that the optimal solution has at least four vertices. First we compute a close-to-optimal solution P for the minimum cost M-connected pinning problem with c(P) :::; ~c*. Since G' = G + K(P) is M-connected, it is 2-connected. Then we compute an optimal solution P' for the minimum cost 3-connected pinning problem on G'. Clearly, c(PI) :::; c". It is also clear that G + K (P U PI) is 3-connected and M-connected. Furthermore, c(P U PI) :::; c(P) + c(P I) :::; ~c* holds. •

We remark that the above methods can be used to design a constant factor approximation algorithm for the corresponding augmentation problem as well, in which the goal is to add a smallest set F of new edges to G such that G + F is globally rigid. We omit the details.

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Acknowledgement. I thank Zsolt Fekete and Bill Jackson for several useful comments.

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A. R. Berg and T. Jordan, A proof of Connelly 's conjecture on 3-connected circuits of the rigidity matroid, J. Combinatorial Theory, Ser. B., Vol. 88 (2003), 77-97 .

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A. R. Berg and T . Jordan, Algorithms for graph rigidity and scene analysis, Proc. 11th Annual European Symposium on Algorithms (ESA) , 2003, (G. Di Battista, U. Zwick, eds.), Springer Lecture Notes in Computer Science 2832 (2003), pp . 7889.

[5] R. Connelly, On generic global rigidity, Applied geometry and discrete mathematics, 147-155, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, Amer . Math. Soc., Providence, RI (1991). [6]

R. Connelly, Generic global rigidity, Discrete Comput. Geom., 33 (2005), no. 4,

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Z. Fekete, Source location with rigidity and tree packing requirements, Operations Research Letters, 34, Issue 6 (2006), pp . 607-612.

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Z. Fekete and T . Jordan, Uniquely localizable networks with few anchors, Proc. 4th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, Budapest (June 2005), pp. 144-148 .

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Z. Fekete and T . Jordan, Uniquely localizable networks with few anchors, Proc. Algosensors, 2006, (S. Nikoletseas and J. D. P. Rolim , eds.) Springer Lecture Notes in Computer Science 4240 (2006), pp. 176-183.

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B. Jackson and T . Jordan, Connected rigidity matroids and unique realizations of graphs, J. Combinatorial Theory Ser B, Vol. 94 (2005), 1-29.

[14] B. Jackson and T. Jordan, The d-dimensional rigidity matroid of sparse graphs, J. Combinatorial Theory Ser B, Vol. 95 (2005), 118-133. [15] B. Jackson and T . Jordan, Graph theoretic techniques in the analysis of uniquely localizable sensor networks, in: Localization algorithms and strategies for wireless sensor networks, G. Mao, B. Fidan (eds.), IGI Global (2009), pp. 146-173 . [16] B. Jackson, T. Jordan, and Z. Szabadka, Globally linked pairs of vertices in equivalent realizations of graphs, Discrete and Computational Geometry, Vol. 35 (2006), 493-512. [17]

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Tibor Jordan Department of Operations Research, Eotvos University, Pazmany Peter setany l1G, 1117 Budapest, Hungary e-mail :

jordan~cs .elte .hu

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 173-212 .

NOISE SENSITIVITY AND CHAOS IN SOCIAL CHOICE THEORY

GIL KALAI*

In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. It turns out that there are two extreme typ es of behavior. On one side, there are social welfare functions, such as the majority rule, that lead to stochastic st ability of the outcome in terms of perturbations of individual preferences. We identify and study a class of social welfare functions that demonstrate an ext remely different type of behavior which is completel y chaotic: they lead to a uniform probability distribution on all possible social preference relations and , for every e > 0, if a small fraction e of individuals cha nge th eir preferences (randomly) the correlation between t he result ing social preferences and the original ones tends to zero as the numb er of individuals in t he society increases. This class includ es natural multi-level majority rules.

1. INTRODUCTION How likely is it that small random mistakes in counting the votes in an election between two candidates will reverse the election's outcome? And if there are three alternatives and the society prefers alternative a to alternative b and alternative b to alternative c how likely is it that a will be preferred to c? We will show that for general social welfare functions these two questions are closely related. In this paper we consider the behavior of general social welfare functions with respect to random uniform voter profiles. Namely, the individual pref"I wish to thank Itai Benjamini, Guy Kindler, Nati Linial , Elchanan Mossel, Motty Perry, Oded Schramm , and Yuval Peres for fruitful discussions. Thi s research was supported by an ISF Bikura grant, by an NSF grant, and by Microsoft , Israe l.

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erences on a set of alternatives are uniformly and independently distributed among all order relations. It turns out that there are two extreme types of behavior: social welfare functions, such as the majority rule, that lead to stochastic stability of the outcome in terms of perturbations of individual preferences and social welfare functions that lead to what we refer to as "social chaos." 1 These two types of behavior in the case of two alternatives have been studied by Benjamini, Kalai and Schramm (1999). A simple game (or voting game) G defined on a set N of players (voters) is described by a function v (called the "payoff function" ,) that assigns to every subset (coalition) S of players the value "I" or "0". We assume that v(0) = 0 and v(N) = 1. A candidate is elected if the set S of voters that voted for him is a winning coalition in G, i.e., if v(S) = 1. We will always assume that the game G is monotone, i.e., that if v(R) = 1 and ReS then v(S) = 1. Recall that a simple game is proper if v(S) + v(N\S) ::; 1 for every coalition S, i.e., if the complement of a winning coalition is a losing one. A simple game G is strong if v(S) + v(N\S) = 1 for every coalition S , i.e., if it is proper and the complement of a losing coalition is a winning one. It is convenient to let N = {I, 2, . . . , n} and regard v as a Boolean function V(Xl' X2, ... ,xn ) where each variable Xi is a Boolean variable: Xi E {O, I}, and thus V(Xl,X2,'" ,xn ) stands for v(S), where S = {i EN : Xi = I}. This notation is used in much of the relevant mathematics and computer science literature. It is also consistent with the basic economic interpretation of the variables Xi as signals the voters receive regarding the superior alternative.

A neutral social welfare function F(A) on a set A of m alternatives based on a strong simple game G is defined as follows: the function F(A) is a map that assigns an asymmetric relation R on the alternatives to every profile of individual preferences. Given a monotone strong simple game G and a set A of alternatives, the society prefers alternative a to alternative b if the set of voters that prefers a to b forms a winning coalition. 1 There are many studies of chaos and chaotic dynamics in economics but they appear to be rather different from this one. In the context of social choice theory, the huge number of "vot ing paradoxes" and the fact that "everything can happen" are often referred to as chaotic phenomena and were studied extensively by Saari ; see, e.g., Saari (2001). Saari's works are concerned mainly with specific voting methods like majority, plurality, Borda's method, and some weighted versions of them. We refer to the property that "anyt hing can happen", as "indet erminacy". Indeterminacy is different from yet related to our notion of social chaos ; see Section 7 and Kalai (2004).

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Remark. The term "neutral" refers to the property that the social welfare function is invariant under permutation of the alternatives. Neutrality requires that the simple games describing the choice between two alternatives be strong, and, in addition, that the same simple game is used for every pair of alternatives.f We study probabilistic properties of social welfare functions for random uniform voter profiles. Thus, to every individual, we assign the same probability (11m!) for each one of the m! order preference relations on the alternatives. Furthermore, the individual preference relations are independent. In other words, if there are n individuals, we assign the same probability for each one of the (rnl)" possible profiles of individual preferences. This is a standard probabilistic model which has been extensively studied in the literature especially in the case of the majority rule (see Gehrlein (1997)). It is worth noting at this point that we will be studying the asymptotic properties of strong simple games and the associated social welfare functions, and it will be convenient to describe our results in terms of sequences (Gkh=1 ,2,... of simple games.i' We can describe our notion of social chaos using several equivalent definitions and we will start with the following simple definition based on the probability of cyclic social preferences when there are three alternatives. Let G be a monotone strong simple game with n players and let F be a neutral social welfare function on three alternatives based on G. Consider a random uniform profile for the individual order preferences between the three alternatives. Let Pcyc(G) be the probability that the social preferences are cyclic. Arrow's theorem asserts that unless G is a dictatorship, Pcyc(G) > O.

Definition 1.1. The sequence (Gk ) of monotone strong simple games leads to social chaos if

(1.1)

lim Pcyc(Gk) = 1/4.

k-+oo

2 The neutral social choice we consider satisfies the basic Pareto and IIA conditions. We do not assume that the social preference is necessarily an order relation. Sometimes they are referred to as "generalized social welfare functions ." 3 For our results we will not require any relation between the games Gk for different values of k. It is useful to think of the results applied to the case where the various Gk'S represent the "same" type of voting rule, such as "simple majority" for different population sizes. But giving a formal definition of a "vot ing rule" that applies to populations of different sizes is not an easy matter and , in any case, is not needed in this paper.

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There are altogether eight possible social asymmetric relations for three alternatives, six of which are order relations and two of which are cyclic. Consider a social welfare function based on a strong monotone simple game G on three alternatives and the uniform distribution on individual order preferences. Symmetry considerations imply that the probabilities of obtaining each of the six order relations as the social preference relation are equal, as are the two probabilities for the two cyclic relations . Therefore, another way of stating relation (1.1) is that for a social welfare function based on (Gk) on three alternatives under the uniform distribution on individual order preferences, for every asymmetric relation R, the probability, for a random voter profile, that the social preferences are described by R tends to 1/8. A result of Gulibaud (see Gehrlein (1997)) asserts that for the social welfare functions on three alternatives based on simple majority, the probability for a cyclic order relation as the number of voters tends to infinity is 1/4 - (3/(2 ' 1r)) . arcsin (1/3) ~ .08744. In other words, if we assume that the voter profile is uniform and observe that the society prefers alternative a to band b to c, our a posteriori probability of the society preferring a to c tends to 0.876. In contrast, in a similar scenario where the sequence Gk satisfies relation (1.1) our a posteriori probability of the society preferring a to c will tend to 0.5 (which is the a priori probabilityj.f We will now move from three alternatives to two and consider the effect of random noise. Suppose that there are two candidates and that the voters' preferences are random and uniform or equivalently that Xi = 1 with probability 1/2 independently for all voters. Noise sensitivity is defined as the effect of random independent mistakes in counting the votes. Formally, for a strong simple game G and t > 0, consider the following scenario: first choose the voter signals Xl, X 2, . . . ,Xn randomly such that Xi = 1 with probability p = 1/2 independently for i = 1,2, ... , n. Let S = V(XI,X2,'" ,xn ) . Next let Yi = Xi with probability 1-t and Yi = I-Xi with probability t, independently for i = 1,2, .. . , n. Let T = V(YI, Y2, . .. ,Yn). Define Nt(G) to be the probability that S =1= T .

Definition 1.2. A sequence (Gk)k=I,2 ,... of strong simple games is asymptotically noise-sensitive if, for every t > 0, (1.2) 4 (Kalai (2002) proved that for every monotone strong simple game this a posteriori probability is always larger than 0.5.)

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Our first theorem shows a surprising relation between noise sensitivity for two alternatives and the probability of Condorcet's paradox for three alternatives. Theorem 1.3. A sequence (Gk) of monotone strong simple games leads to social chaos if and only if it is noise-sensitive.

In proving Theorem 1.3 we obtain a simple formula which, for every strong simple game G, directly relates the probability for Condorcet's paradox and noise sensitivity when the probability for an error is 1/3. Proposition 1.4. For every strong simple game G, (1.3)

We will now define the complementary notion of noise (or stochastic) stability. Definition 1.5. A class Q of strong simple games is uniformly noise-stable if for every s > 0 there is u(s) > 0 such that G E Q,

(1.4) Theorem 1.6. The family of simple majority functions M n is uniformly noise-stable. Moreover, Nu(s) (Mn ) ::; s foru(s) = K·s2 for some constant K.

Our next theorems provide several additional equivalent properties of social chaos which we believe justify this notion. We first note that we can consider slightly larger classes of games. A sequence Gk of monotone proper simple games has a strongly diminishing bias if for a random uniform voter profile, the probability of a "tie" tends to zero as n tends to infinity. An example is the class of simple majority games. All our results extend from strong simple games to the case of monotone proper simple games with strongly diminishing bias. An asymmetric relation R on a finite set X is a binary relation such that every pair of elements x, y E X is ascribed one and only one of the relations xRy or yRx. Clearly, there are 2(';) asymmetric relations on a set of m alternatives. For a monotone strong simple game G and an asymmetric relation R on a set of m alternatives, let PR(G) be the probability under the uniform distribution on voter profiles that the social welfare function based on G will lead to R as the social preference relation.

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Theorem 1.7. Let (G k) be a sequence of proper strong simple games. The following properties Pm are equivalent for m ;::: 3: (Pm): For a social welfare function based on (Gk) on a set A of m alternatives and every asymmetric relation R on A,

(1.5) Note that (P3) is simply our definition of social chaos. It is easy to see that (Pm) implies Pm' when m' < m. Surprisingly, the reverse implication also holds when m' ;::: 3. Consider a neutral social welfare function and a random voter profile, and let R be the resulting preference relation for the society. Consider the following scenario: Suppose that every voter with a small probability t reconsiders and reverses the order of the two alternatives he ranked in the two last places. Let R' be the new social preference relation.

Theorem 1.8. Let m ;::: 2 be a fixed integer and t, 0 < t < 1/2, be a fixed real number. A sequence Gn leads to social chaos if and only if anyone of the following equivalent properties is satisfied:

(Am) The correlation between R and R' tends to zero with n . (Bm ) For every two asymmetric relations R 1 and R 2 on A, the probability that R' infinity.

= R2' conditioned on R = Rl' tends to 1/2(';') as n tends to

Remark. Here we consider Rand R' as probability distributions on asymmetric preference relations, and by talking about the correlation between Rand R' we refer to the correlation between these two distributions. (Of course, B m implies Am, but as it turned out they are both equivalent to social chaos.) (Properties A 2 and B 2 both coincide with the notion of noise sensitivity.) In Section 2 we provide the mathematical tools and proofs for our results concerning the equivalence between the different properties of social chaos. We also describe there relations with the Banzhaf power index and additional characterization in terms of the correlation with weighted majority functions. Benjamini, Kalai and Schramm (1999) showed that the class of weighted majority (strong) games is uniformly noise-stable. A stronger quantitative version was proved by Peres (2004). Benjamini, Kalai

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and Schramm (1999) also proved that if (Gk) is sequence of strong simple games that is asymptotically uncorrelated with weighted majority games, then it is noise-sensitive. In Section 3 we will analyze several examples. Consider the class of multi-level majority-based voting rules described inductively as follows: (i) Simple majority on an odd number of at least three voters has one layer. (ii) Suppose that the players are divided into an odd number of three or more groups and on each group we consider a game with r layers. If G is defined by the simple majority of the outcomes, then G is said to be a multi-layer majority-based simple game with r + 1 layers. Theorem 1.9. A sequence (Gk) of multi-layer voting rules based on simple majority leads to social chaos if and only if the number of layers tends to infinity as k does.

The reason for the chaotic behavior is that any new level of majority will amplify the probability of a cyclic outcome in three alternatives and by a similar argument every new level will amplify the noise introduced by mistakes in counting the votes. When we base our rule on supermajority, social chaos may already appear for two layers. Consider a set of n individuals that is divided into many committees of size k. The decision between two alternatives is based on the majority of committees that prefer one of the alternatives by a 2/3supermajority. This rule defines a proper simple game and in order to have a strongly diminishing bias we require that k = o(logn). We will show in Section 3 that this example leads to complete social chaos when the size of the committees tends to infinity with n. The reason for the chaotic behavior in this case is that the outcomes of the elections are determined by a small number of committees with "decisive views"; i.e., they are sufficiently biased towards one of the alternatives. Such a strong bias in favor of one of the alternatives is unlikely to survive following a small perturbation of the individual preferences. We note that hierarchical or recursive aggregation of preferences similar to those in the first example, and cases where the social preferences heavily depend on a small number of small communities with decisive views, can both represent realistic situations of preference aggregation. The assumption of uniform and independent voter behavior is a standard one and is the basis of a large body of literature. Most of this literature deals with the majority voting rule (see, e.g., Bell (1981) and Gehrlein (1997)). Assuming uniform and independent voter behavior is unrealistic, though it

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can be argued that for issues concerning noise sensitivity and Condorcet's paradox this assumption represents a realistic way of comparing different voting methods. In Sections 4 and 5 we examine our probabilistic assumptions and discuss the situation under more general probability distributions for voter signals/behavior. Section 4 is devoted to the case that either the probability distribution or the voting rule are a priori biased. In Section 5 we present an example of chaotic behavior under the majority voting rule, when voter behavior is not independent. Perhaps the most important form of "noise" in real-life elections and other forms of aggregated choice is abstention. In Section 6 we extend our model to allow for individual indifference between alternatives. Analyzing this model, we find out that noise sensitivity implies that a small change in the fraction of voters who decide to abstain has a dramatic effect on the outcomes , similar to the effect of random mistakes in counting their votes. Section 7 is devoted to a discussion of issues related to noise stability, information aggregation, indeterminacy, the asymptotic nature of our results, and possible extensions to other economic models. Section 8 concludes.

2.

PROOFS OF THE EQUIVALENCE THEOREMS

In this section we prove the various equivalent properties of sequences of simple games that lead to social chaos. We also state additional equivalent formulations. The proofs rely on two methods. A rather simple coupling argument allows us to move from noise sensitivity to the conclusions concerning more than two alternatives given in Theorems 1.3, 1.7, and 1.8. The only remaining step is to show that our definition of social chaos in terms of the probability of Condorcet's paradox for three alternatives implies noise sensitivity. I am not aware of a direct probabilistic argument and the proof of this implication relies on a rather elementary harmonic analysis of Boolean functions. This part of the proof requires mathematical techniques and concepts developed in Benjamini, Kalai and Schramm (1999) and in Kalai (2002). Let (Gk) be a sequence of strong simple games. (All our arguments apply unchanged to the case of proper simple games with strongly diminishing bias.) We consider the following properties:

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• (NS) Noise sensitivity

• (Pm) Asymptotically uniform distribution of social preference relations when there are m alternatives, m 2: 3 • Properties (Am), (B m), m 2: 2. We wish to prove that all these properties are equivalent .

2.1. A coupling argument Proof of the implication (NS) =* (Pk ) . We rely here on a simple (coupling) idea which nonetheless requires careful application. Consider a random voter profile. For two alternatives a and b consider the set V of voters for whom alternatives a and b are consecutive. We will try to determine the effect of reversing the order of the two alternatives a and b with probability 1/2 for all voters in V .

Let Rand R' be two asymmetric relations that differ only for the two alternatives a and b, namely, for a set {c, d} =I- {a, b} of two alternatives cRd if and only if cR'd and for the two alternatives a and b, aRb and bR'a. We will derive from noise sensitivity that (2.1) Consider the following process of determining two profiles P and P' of individual preference relations: a

Step (1): Determine the preference order relations between alternatives and b for every voter randomly and uniformly.

Step (2): Choose the order preference relations for all voters subject to the voters ' preference between a and b that was determined in step (1). Let P be the resulting profile of voters' order relations obtained by this process. Of course, P is a uniformly distributed random profile. Let V be the set of voters for whom alternatives a and b are consecutive in the preference relation. The set V is a random subset of the set of voters . The probability that a voter i belongs to V is l/(m - 1) and these events are independent. Moreover, the set V and the random voter preference relations are independent.

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Let P' be the voter profile obtained from P by reversing the order relations between alternatives a and b for every voter in V . Note that P' is also a uniformly distributed random profile. Let R be the asymmetric relation obtained by applying the social welfare function F to the profile P and let R' be the asymmetric relation obtained by applying F to P' . Note that the two relations Rand R' coincide for every pair of alternatives except possibly a and b. If we focus on these two variables we note that the distribution of voter preference relations between a and b in P is random. We claim that the correlation between the events aRb and aR'b tends to zero as n tends to infinity. (In other words, the probability that aR'b given that aRb tends to 1/2.) Indeed, this follows from the fact that limn --+ oo Nt(Gn ) = 1/2 for t = 1/(m - 1). •

The proof of the implication (NS) =? (B m ) is similar but a little more complicated to the proof presented above. We omit the details. Proof of the implication (Pm) =? (p:n) for m' :::; 3, m' < m, Let A be a set of m alternatives and let A' be a subset of A containing m' alternatives. A random uniform voter profile on A induces a random uniform voter profile on A'. Every asymmetric relation R' on A' can be extended to precisely 2(';')-('~') asymmetric relations R on A. Given such an asymmetric relation R, the probability that a social welfare function based on Gk will lead to R tends to 1/2(';') . (By our assumption (Pm)') Therefore the probability that the restriction on A' will lead to R' tends to 1/2(":;') as required. • The same argument shows the implications: (Am) =? (Am'), for 2 < m' <m. 2.2. An elementary harmonic analysis argument Proof of Theorem 1.3. As mentioned in the Introduction, a simple game can be described by a Boolean junction, namely, j(XI, X2, . . . , x n ) , where the variables Xk take the values a or 1 and the value of j itself is also either a or 1. Every 0-1 vector x = (Xl, X2, . . . , x n ) corresponds to a subset of players S = {k : Xk = I} and we let j(Xl,X2 , . . . , X2) = v(S). In other words, the

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Boolean function f(Xl' X2, . . . , x n ) is just a slightly different notation for the payoff function v(S), and we will refer also to f as the payoff function of the game. Let On denote the set of 0-1 vectors (Xl, ... , xn ) oflength n. Let L2(On) denote the space of real functions on On, endowed with the inner product

(1,g) = The inner product space L2(On) is 2n-dimensional. The L 2-norm of defined by

f is

Ilfll~ = (1,1) = Note that if f is a Boolean function , then j2(x) is either 0 or 1 and therefore Ilfll~ = L:(xI, ...,Xn)EOn 2- n f2(x) is simply the probability that f = 1 (with respect to the uniform probability distribution on On). If the Boolean function f represents a strong simple game then IIfll~ = 1/2. For a subset S of N consider the function

Us (Xl,X2, ,, , ,X n ) -_ (-1 )I:{Xi :i ES } . It is not difficult to verify that the 2n functions Us for all subsets S form an orthonormal basis for the space of real functions on On'

For a function

f

E

L2(nn) , let 1(s) = (1, us)

(1(S) is called a Fourier-Walsh coefficient of 1). Since the functions Us form an orthogonal basis it follows that (2.2)

L

(1,g) =

1(Srff(S).

SeN

In particular,

(2.3)

Ilfll~ =

L

j2(S).

SeN

This last relation is called Parseval's formula.

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Remark. We will demonstrate now the notions introduced here with a simple example. Let fs denote the payoff function of the simple majority game with three players. Thus , fs( xI, X2, X3) = 1 if Xl + X2 + X3 2: 2 and f(XI,X2,X3) = 0, otherwise. The Fourier coefficients of fs are easy to compute : h(0) = 2:(1/8)fs(x) = 1/2. In general if f is a Boolean function then 1(0) is the probability that f(xJ = 1 and when f lepresents the payoff function of a strong simple game f(0) = 1/2. Next, fs ({I}) = 1/8(fs(0, 1, 1) - fs(l, 0,1) - fs(l, 1,0) - fs(l, 1, 1)) = (1 - 3)/8 and thus

h( {j}) =

h(8) = 0 when 181 = 2 and finally + fs(l, 0,1) + fs(O , 1, 1) - f(l , 1,1)) = 1/4.

-1/4, for j = 1,2,3. Next,

h ({I, 2, 3}) = 1/8(fs(1 , 1,0)

We point out the following simple result: Proposition 2.1. If a Boolean function f is the payoff function of a strong simple game then 1(8) = 0 whenever 181 is an even positive integer. Proof. Let 9 = f -1/2 (= f(x) -1/2u0)' Since f represents a strong simple game g(l - XI, 1 - X2 , ... , 1 - x n ) = -g(XI' X2 , ... , x n ) . When 181 is even, consider the contributions of (Xl, X2 , . .. , x n ) and (1- Xl , 1- X2 , . . . , 1- x n ) to the expression g(8) = (g, us). Note that these two contributions cancel out and therefore g(8) = 0 for every set 8 of even size. It follows that when 8 is a nonempty set of even size, 1(8) = g(8) = O. • Let f be a Boolean function. For k 2: 0, let

(2.4) Theorem 2.2. The probability Pcyc(G) of a cyclic outcome of a social welfare function on three alternatives based on a strong simple game G with a payoff function f is n

(2.5)

Pcyc(G) = 1/4 -

L (l/3)k-I Wk(f ). k=l

Theorem 2.2 is from Kalai (2002). Theorem 2.3. Let G be a strong simple game and let f be its payoff function. The probability that an e-tioise will change the outcome of an election is given by the formula: n

(2.6)

Ne(G) = 1/2 - 2 ·

L (1- 2c)kWk(f ). k=l

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Theorem 2.3 is from Benjamini, Kalai and Schramm (1999). The proofs of Theorems 2.2 and 2.3 are quite elementary and rely essentially on relations (2.2) and (2.3). Note that we obtain the simple relation (Proposition 1.4) between the probability for Condorcet's paradox and noise sensitivity when the probability for a noisy bit is 1/3.

(2.7) From Theorem 2.3 we can easily derive (see Benjamini, Kalai and Schramm (1999)):

Corollary 2.4. A sequence (fn) of Boolean functions that represent strong simple games is noise-sensitive if and only if for every k > 0 (2.8)

As a matter of fact Corollary 2.4 holds as stated when the Boolean functions (fn) do not necessarily represent strong simple games but rather satisfy that the sequence of probabilities that in = 1 is bounded away from 0 and 1 as n tends to infinity.

Proof of Theorem 1.3. For this proof we need to compare the information given by Theorem 2.2 and Corollary 2.4. Suppose that Pcyc(G n) -7 1/4. Let in be the payoff functions of the strong simple game Gn. It follows from relation (2.5) that for every r > 0, limn-too Wr(fn) = 0, which by Corollary 2.4 is equivalent to noise sensitivity. On the other hand, if (fn) satisfies relation (2.8) then it follows from relation (2.5) that limn-too Pcyc(Gn) = 1/4. (This implication was also proved earlier.) _ We now complete the proof of Theorems 1.7 and 1.8. We have already proved the following implications: (P3) ~ (NS) ~ (Pm) ~ (P3) and (NS) ~ (B m), for every m ~ 3. Of course, if property (Pm) holds then property (Am) is equivalent to property (B m) and therefore (NS) ~ (Bm), m ~ 3. On the other hand , (B m) ~ (B 2 ) = (NS) and (Am) ~ (A 2 ) =

(NS) . Remark. We now continue to demonstrate the notions and results introduced here with our simple example. As before, h denote the payoff function of the simple majority game with three players. As we saw, the

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Fourier coefficients of fa are described by h(0) j = 1,2,3, h(8) = 0 when

181 =

2 and

= 1/2, h( {j}) = -1/4, for

h({I, 2, 3}) =

1/4. When there are

three alternatives the probability for cyclic social preferences given by formula (2.5) is 1/4 - 3· (1/4)2 - (1/9)· (1/4)2 = 8/(9 ·16) = 1/18. This value agrees with the well-known outcome given by a direct computation. The value of N t(f3) is 1/2-2·(1-2t)(3/16)-2·(1- 2t)3.(1/16) = 3t/2-3/2t 2+t3. (Again, this can be derived by a direct computation.)

2.3. Individual power We conclude this section with a short discussion of the relation between noise sensitivity and the Banzhaf power index. Let bj (G) denote the Banzhaf power index of player j in a strong simple game G. Recall that bj(G) is defined as the probability that player j is pivotal , i.e., the probability that for a random coalition 8 not containing j it is the case that 8 is a losing coalition but 8 U {j} is a winning one. Let bmax(G) denote the maximum value of the Banzhaf power indices and let I(G) denote the sum of the Banzhaf power indices for all players in a simple game, known also as the total influence of the game. It is known that in the case of n players, I(G) is maximized by a simple majority game. (See, e.g., Friedgut and Kalai (1996).) For simple majority games I(G) is proportional to ..;n. Definition 2.5. The sequence (Gk) of monotone simple games has a diminishing individual Banzaf power if

Proposition 2.6. If the sequence (G k ) of monotone simple games leads to social chaos then it has a diminishing individual Banzhaf power. The proof is immediate: when there is a player with a substantial Banzhaf power the outcome of the game has a substantial correlation with this player vote. A small noise in counting the votes will miss the vote of this player with a substantial probability and therefore the correlation of the outcomes before and after the noise will also be substantial. (Here "substantial" means bounded away from zero.) Definition 2.7. The sequence (Gk) of monotone simple games has a bounded power ratio if the ratio bmax(Gk)/bmin(Gk) is uniformly bounded.

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Given a monotone simple game G, let p(G) be the probability that a random uniform subset of players is a winning coalition. If G is strong, then p(G) = 1/2. Given two monotone simple games G and H on the same set N of players let p(G, H) be the probability that a random uniform subset S of players is a winning coalition for both G and H . The correlation between G and H, denoted by cor (G, H), is defined by cor (G, H) = p(G, H) - p(G)p(H). The well-known FKG inequality asserts that for every two monotone simple games cor(G, H ) ~ O. Definition 2.8. Two sequences (Gk) and (Hk) of monotone strong simple games are asymptotically uncorrelated if

lim cor (Gn , H n ) = O.

n-+oo

Weighted majority games are important class of simple games. Given n nonnegative weights WI , W2, • . . , W n , and a value W , define the payoff function v(S) to be one if L:iES Wi > Wand v(S) = 0, otherwise. (If W = (1/2) L:~I Wi and no partial sum of weights is equal to W the weighted majority game is strong.) Theorem 2.9. A sequence (Gk ) of monotone strong simple games leads to social chaos if and only if it is asymptotically un correlated with every sequence of weighted majority games.

An important special case is: Theorem 2.10. A sequence (Gk) with a bounded Banzhaf power ratio leads to social chaos if and only if it is asymptotically un correlated with the sequence of simple majority games.

Theorem 2.10 together with our earlier results demonstrates the "robustness of majority":5 Voting methods (based on proper simple games where the voters have comparable power and the probability of a tie tends to zero) either have a substantial correlation with simple majority or else lead to social chaos. Theorem 2.11. Let (Gn ) be a sequence of monotone strong simple games with a bounded power ratio and suppose that Gn has n players. The sequence Gn leads to social chaos if and only if limn -+ oo I(G n ) / yin = o. 5

A term coined by Dasgupta and Maskin in another cont ext .

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This result can be derived directly from the following result from Benjamini, Kalai and Schramm (1999): Theorem 2.12. The sequence (Gk ) is noise-sensitive if and only if

(Here the sum extends over all players of the game Gk . ) Remark. Holzman, Lehrer and Linial (1988) proved that for every monotone simple game (bk(G)) 2 ::; 1.

z=

3.

MULTI-LEVEL MAJORITY

3.1. Simple majority Our principal examples of social welfare functions that exhibit social chaos are based on multi-level majority. For their analysis we will have to start with the case of simple majority. The outcomes of simple majority under random individual voter profiles have been the subject of intense study. We will first require a few results concerning the behavior of simple majority. Given a monotone strong simple game G recall that Pcyc(G) denotes the probability that for a social welfare function on three alternatives A, Band C and a random uniform voter profile the social preferences will be cyclic. We start with a simple result concerning the majority voting rule with an odd number of voters. Lemma 3.1. Let M n be a simple majority game on an odd number n of players, n 2: 3. Then, (i) Nf;(M n ) 2: Nf;(M3 ) = 3/2 · e - 3/2£2 + £3;

(ii) Pcyc(Mn) 2: Pcyc(M3 ) = 1/18. The proof is given in Section 9.1 in the Appendix. We point out that Fishburn, Gehrlein and Maskin proved that Pcyc(M2k+l ) 2: Pcyc(M2k - l ) for every k ~ 3. This implies, in particular, part (ii) of Lemma 3.1. The next result describes the limiting behavior of majority when the number of voters tends to infinity.

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Proposition 3.2. (1) (Sheppard (1899)) For every 6 > 0 · N (M ) _ arccos (1- 26) 11m e n t

(3.1)

>:

n-+oo

•

1r

(2) Gulibaud (see Gehrlein (1997)) lim Pcyc{Mn) = 1/4 - (3/{2'1r)) . arcsin (1/3) ~ .08744.

n-+oo

Proposition 1.4 gives a direct link between Gulibaud's formula and Sheppard's formula. Relation (2.7) asserts that for every n, Pcyc{Mn) = 3/2N1/ 3{Mn ) - 1/2. Together with the case 6 = 1/3 of Sheppard's formula limn -+ oo N 1/ 3{Mn ) =

arcco:(1/3)

we obtain the formula of Gulibaud.

It follows from relation (3.1) that as 6 tends to zero limn -+ oo Nc{Mn) = (1 + 0(1)) . v'2i/1r. This is a more precise version of Theorem 1.6. We will not reproduce here the proofs of Proposition 3.2 but rather will explain informally why, for simple majority on a large number n of players and for a small amount 6 of noise, the function Nc{Mn) behaves like ..ft. The median gap between the number of votes of the winning candidate and the losing one in a simple majority is close to yri. According to the central limit theorem the probability that the gap is larger than tyri behaves like e- t 2 / 4 • It follows that a large gap is extremely rare while below the median gap the distribution of the gap is rather close to uniform. When we flip eti random votes the median gap between the number of votes for Alice that now go to Bob and votes for Bob that now go to Alice is thus close to ..ftyri, and again it is rare that this gap is much larger. In order for the noise to change the election 's results we require (apart from an event with a small probability) that the original gap between the votes for Alice and Bob be around ..ftyri and this occurs with probability of roughly ..ft. On the other hand, if the gap between votes for Alice and votes for Bob behaves like ..ftyri then the probability that the noise will change the election's outcome is bounded away from zero. This argument indeed shows that Nc{Mn) behaves like ..ft.

3.2. Multi-level games based on simple majority The first class of examples that exhibit social chaos are based on multi-level majority where the number of levels increases. Recall the class of multilevel majority-based voting rules described inductively in the Introduction: (i) Simple majority on an odd number of at least three have one layer.

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(ii) Suppose that the players are divided into an odd number of three or more groups and on each group we consider a game with r layers. If G is defined by the simple majority of the outcomes then G is said to be a multi-layer majority-based simple game with r + 1 layers. Define u(1)(e) = u(e) = 3/2· e - 2e2 + 5/4e3 and u(r) (e) = u(u(r-l)(e)), r = 2,3,4, . . . . Theorem 3.3. Let G; be an r-layer simple game based on simple majority. Then As e tends to 0,

Proof. The proof is by induction on r and the case r = 1 is given by Lemma 3.1. Consider a random vector (XI,X2, ... ,xn ) E Dn . The value of V(XI,X2, •.. ,xn ) is the simple majority of the (r - 1) level outcomes Zl, Z2,.··, Zm. Suppose that we flip each variable Xi with probability e. By the induction hypothesis each Zi is flipped with probability of at least u(r-l)(e) and by Lemma 3.1 the probability that v is flipped is at least u(u(r-l))(e) = u(r) (e). When G = T; is the recursive ternary majority game with r levels we obtain that Nc(Tr ) = u(r)(e). To prove Theorem 3.3 we need to verify that for every e > 0, limr-too u(r)(e) = 1/2. We will state separately the result for a bounded number of levels. Theorem 3.4. Let ¢(e) = ¢(1)(e) = arcco~(1-c) and let ¢(i+l) (e) = ¢(¢(i) (e), i = 1,2, . . .. Let r be a fixed positive integer. Consider a sequence (Gn ) of r-level majority strong simple games where each level is based on simple majority of i« players. (Thus, n = t~.) Then for a fixed e > 0 as n tends to infinity,

As e tends to zero

In our model, simple majority is considerably more stable in the presence of noise than a two-layer election method like the U.S. electoral system. In an electoral system with many states and many voters in each state, Nc(G) is proportional, for small values of e, to e 1/ 4 . In a three-stage system of this kind Nc(G) behaves like e 1/ 8 . Is the advantage of simple majority

Noise Sensitivity and Chaos in Social Choice Theory

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on a two-level majority in terms of noise sensitivity relevant for evaluating the U.S. electoral method? Can we conclude that events like those in the 2000 U.S. presidential election are less likely to occur in a simple majority system? While the probabilistic models are unrealistic, it appears that the advantage of simple majority compared to a two-level electoral method is quite general. Remark. The hierarchical voting methods considered here resemble the multi-tier system of councils ("soviets" in Russian). Lenin (among others) advocated this system during the 1917 Russian Revolution. Lenin's concept of centralized democracy is based on a hierarchical method of voting and was implemented in the Soviet Union and its satellites for party institutions, national bodies, and labor unions. (For national bodies, the method was changed in 1936.) For party institutions (which were the most important) there could be as many as seven levels. Party members in a local organization , for example, the Department of Mathematics in Budapest, elected representatives to the Science Faculty party committee who in turn elected representatives to the University council. The next levels were the council of the 5th District of Budapest, the Budapest council, the Party Congress, the Central Committee and finally the Politburo." Friedgut (1979) is a good source on the early writings of Marx, Lenin, and others , and for an analysis of the Soviet election systems that were prevalent in the 70's.

3.3. Two-level games based on supermajority

The second class of examples of social welfare functions that exhibit social chaos are based on a two-level method where the lower level is biased. Consider a two-level proper game G = G[a, b, t], where t satisfies b/2 < t < b, defined as follows. There are a . b voters divided into a communities of b voters each. Given a subset 8 of voters, we call a community C positive if 18 n CI ~ t and negative if 18 n CI :s; b - t. The subset 8 is a winning coalition if there are more positive communities than negative communities . Let Gn = G[a n, bn,tnJ be a sequence of such games. In order for Gn to have strongly diminishing bias it is necessary and sufficient that the expected number of decisive coalitions tends to infinity. 6

I am thankful to Lad for this example.

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Theorem 3.5. If Gn = G[an, bn,tnl has strongly diminishing bias and lim bn =

n-too

00,

then the sequence Gn leads to social chaos. The proof is based on a straightforward but tedious computation and is omitted.

Remark. It can be shown by similar proofs to those we present that social chaos occurs for more general hierarchical voting methods. This applies both in the case where in each level the method is balanced and the number of levels tends to infinity and in the case where there are as little as two levels and the method is biased in the lower level. Social chaos may occur also for more complicated recursive methods. For example the different communities that we considered need not be disjoint. But the analysis in such cases can be considerably harder.

4.

BIAS

The framework described in this paper extends to more general probabilistic distributions of voter behavior (or individual signals) . We expect that the dichotomy described here between stochastically stable and chaotic behavior can also be extended, though such extensions are conceptually and mathematically more difficult. When we consider the effect of stochastic perturbations there are also various possible models for the "noise." Does it represent mistakes in counting the votes? or, perhaps, a small fraction of voters reconsidering their positions or considering whether or not to vote? We also expect that the stochastic stability of the majority rule extends to various other probability distributions representing the voters' behavior . Our probabilistic model assumes three properties which we will discuss separately: (1) Independence: Voters' probabilities (or signals) are independent. (2) No bias: For two alternatives a and b the probability that the society prefers alternative a to alternative b is 1/2. (3) Identical voters: Voters' probabilities (or signals) are identical.

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The assumption of identical voter preference distribution is unrealistic but I expect that the basic results of this paper can be extended to cases where the assumption of identical voter behavior is relaxed while the other two assumptions are kept . We will concentrate on the other two assumptions and discuss in this section bias and in the next section dependence between voters. The assumption of no bias is clearly an ideal assumption. A more realistic assumption in many cases of aggregated choice is the assumption of "small bias," which is defined as follows:

Definition 4.1. Consider a sequence (Gk, Vk), where Gk is a proper simple game with nk players and v» is a probability distribution on coalitions of Gi; (Alternatively, Vk can be regarded as a probability distribution on 0-1 vectors of length nk.) The sequence (Gk, Vk) has small bias if for some constant t > 0, the probability PVk (Gk) that a random coalition according to Vk is a winning coalition satisfies

While it is hard to justify that a priori two candidates in an election have the same probability of winning it is often quite realistic that the a priori probability of winning is substantial for both. We will return to this issue towards the end of this section . We will describe in this section the extension of our results to cases with bias. Bias occurs when the simple game is strong and thus neutral between the candidates but the probability distribution of signals favors one of the candidates, and also when the probability distribution is uniform but the voting method itself is described by a simple game giving advantage to one of the candidates. The results in these two types of extensions (and their combination) are similar . The dichotomy between noise-stable and noisesensitive monotone simple games and many of the results presented in this paper can be extended to the case of independent voter behavior with a small bias. Recall that nn denote the set of 0-1 vectors (Xl, X2, . . . , x n ) oflength n. Given a real number P, 0 < P < 1, P p will denote that product distribution on nn defined by

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where k = function v

Xl

+ X2 + ...+ X n .

For a monotone simple game G with payoff

= V(XI, X2 , " . ,xn ) , define

Let t > 0 and consider the following scenario: first choose the voters' signals Xl, X2, .. . ,Xn at random such that Xi = 1 with probability p independently for i = 1,2 , . . . , ti. Consider S = V(XI' X2, . . . , x n ) . Next let Yi = Xi with probability 1 - t and Yi = 1 - Xi with probability t, independently for i = 1,2, ... , n. Let T = V(YI, Y2," " Yn). Let Cf(G) be the correlation between Sand R. Definition 4.2. Consider a sequence (Gk,Pk)k=I,2, ... where Gk is a monotone simple game and 0 < Pk < 1. The sequence (Gk,Pk) is asymptotically noise-sensitive, if for every t > 0, lim Cfk(G k ) = O. k-too In the case of small bias, namely, when Ppk(Gk) is bounded away from o and 1, Theorem 1.7, which asserts that simple majority is noise-stable , extends and so do Theorems 2.9 and 2.12. If limk-too P Pk (Gk) = 1 (or 0) the sequence is always noise-sensitive.

Theorem 4.3. Every sequence (Gk,pk) of monotone simple games where limn-t oo Ppk(Gn ) = 1 (or 0) is noise-sensitive . Theorem 4.3 follows from a Fourier-theoretic interpretation of noise sensitivity, which extends the formula for the uniform distribution, and a certain inequality of Bonami and Beckner. Perhaps some explanation of the meaning of noise sensitivity in the presence of bias is in order . When there are two alternatives noise sensitivity can be described in two ways: (i) The result following a small perturbation is asymptotically uncorrelated with the original result.

(ii) The probability that a small perturbation will change the collective choice tends to 1/2.

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The reason that when there is no bias (i) implies (ii) is that the model assigns equal probabilities to the two outcomes. Property (i) may extend to rather general probability distributions but the dramatic conclusion in (ii) requires that there be no bias. Assuming small bias, condition (i) implies the following: (iii) With a probability uniformly bounded away from zero, a small amount of noise will change the outcome of the election . Note that for a sequence of games (Gk) with small bias in order to conclude property (iii) it is enough to assume a weaker property than noise sensitivity. Property (iii) follows if the sequence (Gk ) , in addition to having small bias, is uniformly chaotic, which is defined as follows:

Definition 4.4. Consider a sequence (Gk,Pkh=1,2,... where Gk is a monotone proper simple game and 0 < Pk < 1. The sequence (Gk,pk) is uniformly chaotic, if there is 0: > 0 so that for every t > 0, lim Cfk(Gk)

k-+oo

< 1- 0: .

It is interesting to compare what happens to conditions (i) and (iii) when the bias get larger (maintaining the assumption of independent and identical voter preferences). In an election between Alice and Bob, when the probability of Alice winning tends to zero then so does the probability that a small random perturbation in voter behavior will change the outcome of the election. On the other hand, Theorem 4.3 asserts that in this case it is always true, even for the majority rule, that the correlation between outcomes before and after the noise also tends to zero. Roughly speaking, Theorem 4.3 asserts that when your probability of winning the election is small and you have the option of forcing a recount of a random fixed fraction of votes, this option almost doubles your chances of winning (for the majority rule and for any other voting rule). Our next proposition, whose proof follows directly from the definition of uniform noise stability, asserts that the property of noise stability is itself stable under small changes in the probability distribution. Let n(k) be a monotone sequence of positive integers. Consider two sequences (Ilk) and (110 where Ilk and II~ are probability distributions on !1 n (k)' We say that II and II' are asymptotically non-singular if for every sequence of events Sk E !1 n(k), limll(Sk) = 0 if and only if limll'(Sk) = O.

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Proposition 4.5. Consider a family g of strong simple games which is uniformly noise-stable. Suppose that Pn is a sequence of probabilities such that the distributions P pn on nn are asymptotically non-singular to the uniform distribution. Then g is uniformly noise-stable w.r. t. the distributions P pn' The stability of uniform noise stability extends to small perturbations of the voters' profile distribution, and the distribution describing the noise does not require the assumption of independent voters. We will now briefly deal with three or more alternatives. Let", be a probability distribution on order relations on a set A of alternatives. Given a set of n voters we will denote by P 71 the probability distribution on the voters' profile where the voters' preferences are independent and are described by n, We will assume that for every two alternatives a and b the probability that aRb when R is chosen at random according to ", is strictly between 0 and 1. We denote this probability by ",(a, b). We will study the distribution of social preferences when the voters' profiles are distributed according to P 71' Consider the probability distribution PTJ defined on asymmetric relations as follows:

(4.1)

pTJ(R) =

II ",(a , b) / L II ",(a, b). aRb

R

aRb

This distribution PTJ(R) represents the following scenario. For every individual the order relations among the m alternatives are distributed according to ", and are independent. The distribution on the social preference is based on the following procedure. For every two alternatives a and b choose an individual Va ,b and let the the social preference between a and b agree with the preference of this individual. Assume that all these individuals are different . The resulting distribution on the social preferences is PTJ' We will now extend the definition of social chaos to strong monotone simple games and arbitrary distributions", on the individual order relations.

Definition 4.6. A sequence (Gkh=1,2,... of strong simple games leads to social chaos w.r .t . the probability distribution", on voters' order preferences if for every three alternatives a, b, and c the probability that the society prefers alternative a to alternative c given that it prefers a to band b to c is asymptotically the same as the a priori probability that the society prefers a to c.

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Given these definitions, Theorems 1.3, 1.8, 1.7 extend. We need to replace the uniform distribution on all asymmetric relations by the distribution PTJk' We conclude this section by a brief discussion of the assumption of small bias. The assumption of small bias is realistic and this by itself can be regarded as a surprising phenomenon. Why is it the case that so often shortly before an election we can give substantial probability for each one of two candidates to be elected? How come the probabilities that we can assign to the choices of each voter do not "sum up" to a decisive collective outcome? This seems especially surprising in view of the property of aggregation of information. It is worth noting that there are several factors that can push the situation of collective choice towards small bias: (i) Abstention and other strategic considerations of voters: If a voter's incentive to vote is based (as is often assumed) on his chances of being pivotal, then for voters acting strategically, abstention or applying mixed strategies and not acting just based on the signals may drive the situation towards criticality. See, e.g., Feddersen and Pesendorfer (1996, 1998) where supermajority is studied, and Samet (2004) for general simple games ."

If there are two main candidates and a few additional candidates who are each ideologically close to one of these two, a voter may choose to vote for a minor candidate if he knows that his vote is not needed to decide between the main candidates. Again, this can push the situation between the two main candidates towards criticality. (ii) Strategic considerations of candidates: In elections, such as those in the U.S., the candidates concentrate their efforts on swing states. This may lead to critical behavior in some of these states.

(iii) Choice out of equilibrium: Consider the difference between the following two scenarios: a) choosing between two types of products, b) choosing between two types of products when the difference in quality is reflected in their prices. In the second scenario it is more likely that we will observe critical behavior and even more so when the decision-maker's information about the difference in quality is based primarily on the difference in price . (iv) Dependence: A major reason why the assumption of small bias is realistic is the lack of independence between voter preferences. Even a short 7 In th ese cases strategic behavior leads to asymptotic complete aggregation of information in the sense that if there are independent individual signals weakly biased toward the superior alternative then with probability tending to one the superior alternative will be chosen.

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time before an election there is a nonnegligible probability that an event will influence a large number of voters in the direction of one candidate.f

5.

DEPENDENCE

Following is an example that demonstrates some of the issues that arise when the voting rule is simple majority and voter behavior is not independent. Suppose that the society is divided into a communities of b voters each. The number of voters is thus n = ab, which we assume is an odd number. Each voter i receives an independent signal 8i, where 8i = 1 with probability 1/2 and 8i = 0 with probability 1/2. The voters are aware of the signals of the other voters in their community and are influenced by them. Let q > 0 be a small real number. A voter changes his mind if he observes a decisive advantage for the other candidate in his community, i.e., if he observes an advantage where the probability of observing such an advantage or a larger one, when voter behavior is independent and uniform, is at most q. The election's outcome as a function of the original signals 81,82, ... ,8n can be described by a strong simple game which we denote by G[a;b; q]. (The example will also "work" if a fraction (1 - a) of the voters vote according to their signals and only some small fraction a > 0 of voters change their minds as before.f Let us examine the situation for a sequence (Gn ) of such examples where the parameters a and b both tend to infinity, n = ab, q tends to zero, and = o( vn)· (For example, take a = b = and q = n- 1/ 4 . ) In this case, Gn exhibit noise sensitivity for (independent) small amounts of noise in the original signals . The outcomes of elections as a function of the individual signals is thus completely chaotic. Indeed, the outcomes of the elections are determined (with very high probability) by the number of communities with decisive signals in favor of one of the candidates. The expected number of decisive communities is qa. The difference between the number of decisive communities favoring the two candidates behaves like..;qa. Therefore, these

J17CJ

vn,

8 This type of dependence can imply failure of aggregation of information in the following sense: it is possible that every voter's signal will favor a certain candidate with probability (say) 0.6 and yet there is a substantial probability that the other candidate will be elected regardless of the number of voters. 9 Another variant that will have similar properties is the case in which there are election polls in each community that influence some small fraction of voters.

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communities contribute b.;qa votes to one of the candidates. Since the original gap in the number of favorable signals between the two candidates behaves like .J(J) and by our assumption yI(i,n = o(n.;qa) we conclude that the decisive communities determine the outcome of the elections with very high probability. From this point on, the analysis is similar to the proof of Theorem 3.5.10 On the other hand , this same sequence is extremely noise-stable for independent noise with respect to counting the votes! The gap between votes cast for the two candidates behaves like b.;qa so that even if a random subset of 40% of the votes are miscounted the probability that the election's outcome will be reversed is extremely small. The two properties of our example: The first property of chaotic behavior for noise affecting the original signal and the second property of strong stochastic stability for noise affecting individual votes seem characteristic to situations in which voter behavior depends on independent signals in a way that creates positive correlation between voters . This topic deserves further study. Note that when we consider random independent noise in the original signals, the distribution of the resulting votes is identical to the original distribution without the noise. This is not the case for random independent noise in counting the votes. We now move from examples to more general models. A general model for voter behavior will be of a pair (G, v) where G is a strong simple game representing a voting rule and v is a probability distribution of voters' preferences or signals. In this case we can consider random independent noise that can represent mistakes in counting the votes. For such a form of noise the distribution of the noisy signals will be different than the original distribution. Another form of noise can be described as follows. Denote by d(x, y) the Hammming distance between two vectors x, Y E On' Given a vector (Xl,'" , X n ) of voters ' signals and another vector Y = (Yl ,"" Yn) define

q(y) = (1- tt-d(x,y)td(x,y) . v(y), and normalize q(y) to a probability distribution by setting

p(y) = q(y)/

va,

L {q(z) : z E On} .

10 If q = tf where t is small but bounded away from zero , then for every amount e of noise in the signals, the correlation between the outcomes before and aft er the noise will t end to a small const ant ; in this case t he situation depends on the original signals in a very, but not completely, chaotic manner .

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If the noise changes x to y with probability p(y) the distribution of the noisy voters' signals is the same as the original distribution. This type of noise can be regarded as applying to the (unknown) mechanism leading to the voters' preferences distribution t/,

A somewhat more restrictive but still quite general probabilistic setting for aggregation is the following: there are some independent signals 81,82, ... .s, that may depend on the state of the world. At the stage where individuals make up their mind they are exposed to some of these signals directly and also to the emerging preferences of other individuals. The individual's final preferences are then aggregated according to some voting rule. When we study the sensitivity to noise it can make a big difference if we consider noise affecting the original signals or noise affecting the individual choices (mistakes in counting the votes). In such a general framework there are cases, such as the example presented at the beginning of this section , when rather general simple games describe the situation even for a standard voting rule. For simplicity we can restrict ourselves to the case of simple majority rule . Suppose that the choice of the i-th individual Vi = Vi(81,"" 8t) is a function of the signals 81, . . . ,8t. If the Vi'S are monotone and satisfy vi(1- 81 ,1- 82,··.,1- 8t} = 1-Vi (81, 82, ... , 8t) then the outcome of the elections in terms of 81, 82, ... , 8t is expressed by a proper simple game G (see also Kalai (2004), Section 3). If the signals are random and uniformly distributed on S1 t then the situation can be considered within our frameworks. As we have shown, it is possible to provide natural examples in which the elections outcome as a function of the original signals will exhibit chaotic behavior . In addition to examples that are hierarchical, like those we have considered, one can also consider cases in which voters repeatedly update their positions based on the positions of friends and neighbors .

6. ABSTENTION In real elections it is common for a large proportion of voters not to vote. Abstention is thus the most common form of noise when it comes to real-life elections and perhaps also other forms of aggregation. It is possible to extend our model and consider monotone and neutral social welfare functions that allow individual indifference between alternatives. When there are two alternatives a and b, a social welfare function can

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201

be described by a function V(Xl,XZ, .. . ,xn) where each Xi E {-1,0,1} and also v(xl, Xz, ... , x n ) E {-I, 0,1}. Here Xi = 1,0, -1 according to whether the i-th individual prefers a to b or is indifferent between the two, or prefers b to a, respectively. For two alternatives neutrality is equivalent to v( -Xl, -Xz, . . . , -X n) = -V(Xl, Xz, . . . , xn ). When we consider only ±1 variables, the function v describes a proper simple game G. n We assume that v is monotone, namely, if Xi 2 Yi, i = 1,2, . .. , n then V(Xl, Xz, .. . , x n ) 2 V(Yl, in , .. . ,Yn). The function v determines a neutral social welfare function on every set A of m alternatives. We can study the stochastic behavior of social preferences as we did before. Let (vn ) be a sequence of functions of the types we considered above and let (Gk) be the proper simple game associated to (Vk) ' We need the assumption that for every a > if a random fraction of a among voters are indifferent between the two alternatives (in other words, if they abstain) and the voters that do not abstain vote randomly and uniformly, then the probability of social indifference tends to zero as n tends to infinity. This implies, in particular, that the sequence (Gk ) has strongly diminishing bias.

°

Suppose that all voters vote ; then we are back in the case we studied in this paper (with the assumption of strongly diminishing bias) and the notion of noise sensitivity and social chaos extend unchanged.

Our first theorem asserts that if (Gk) is a noise-sensitive sequence of proper monotone games then the effect of abstention of even a small fraction of voters is dramatic. (Note that the game Gk does not determine the function Vk but the result applies simultaneously for all neutral monotone extensions of the games Gk to cases of individual indifference.) Consider the following scenario. Let R be the outcome of an election between two candidates and random voter profile (Xl, Xz, .. . , x n ) without abstention. Now consider another random voter profile (Yl, in , . . . ,Yn) where Yi = Xi with probability 1 - sand Yi = 0 with probability s . Let R' be the outcome of the election for the voter profile (Yl , Yz,· . . , Yn). Theorem 6.1. Consider a sequence (Vk) of monotone functions describing a neutral voting rule with individual indifference. Suppose that the social indifference is strongly diminishing and that the sequence (G k) of associated proper games is noise-sensitive. Then the correlation between Rand R' tends to zero as k tends to infinity. 11 Not e that the payoff function v uses the values -1 , +1 instead of 0, 1 to describe this game .

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Remark. As before, when we refer to "the correlation between Rand R'" we mean "the correlation between the two events": 1.

R represents the social preferences for (Xl, . . . , X n )

2. R' represents the social preferences for (YI, .. . , Yn). We abuse notation similarly below. Our next result asserts that the property of noise sensitivity is preserved with probability 1 under abstention. Next consider the following scenario . Let R be the outcome of an election between two candidates and random voter profile (Xl , X2 , . . . ,xn ) conditional on the assumption that the probability of a voter abstaining is Q 2 o. Let (YI' Y2, .. ·, Yn) be another random voter profile where Yi = Xi with probability 1 - sand Yi = 0 with probability s. Let R' be the outcome of the election for the voter profile (YI ' Y2 , · · · , Yn).

Theorem 6.2. Consider a sequence (Vk) of functions describing a monotone voting rule with individual indifference. Suppose that the social indifference is strongly diminishing and that the sequence (Gk) of associated proper games is noise-sensitive. Then the correlation between Rand R' tends to zero as k tends to infinity. The proofs of these results are omitted.

7.

DISCUSSION

1. The superiority of majority. What are the simple games most stable under noise? It was conjectured by several authors that under several conditions which exclude individual voters having a large power, majority is (asymptotically) most stable to noise. (See a discussion and a surprising application to computer science in Khot, Kindler, Mossel and Ryan (2004)) . This conjecture was recently proved by Mossel, O'Donnell and Oleszkiewicz

(2005) . Recall that The sequence (G k) of monotone simple games has diminishing Banzhaf individual power if

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Theorem 7.1 (Mossel, 0 'Donnell and Oleszkiewicz (2005)). For a sequence (G n ) of strong simple games with diminishing individual Banzhaf power,

In other words, when the individual Banzhaf power indices diminish simple majority is asymptotically most stable to noise.12 A consequence of this theorem is

Corollary 7.2 (Mossel, O'Donnell and Oleszkiewicz (2005)). For every sequence (G n ) of strong simple games with diminishing individual Banzhaf power, Pcyc(Gn) ~ 1/4- (3/(2 '71")) . arcsin (1/3) +0(1) . In other words, for social welfare functions with diminishing Banzhaf value, simple majority is asymptotically least likely to yield a cyclic social preference relation on three alternatives.

2. Aggregation of information. Given a strong monotone simple game G on a set V of players we denote by P p( G) the probability that a random set S is a winning coalition when for every player v E V the probability that v E S is p, independently for all players. Condorcet's Jury theorem asserts that for the sequence Gn of majority games on n players

(7.1)

lim Pp(G n ) = 1,

n-+oo

for every

p> 1/2.

This result, a direct consequence of the law of large numbers, is referred to as asymptotically complete aggregation of information. In Kalai (2004) it was proved that for a sequence of monotone simple games asymptotically complete aggregation of information is equivalent to diminishing ShapleyShubik individual power. The speed of aggregation of information for simple majority games is described by the central limit theorem. For a monotone strong simple game G, let T£(G) = [PI,P2] and te:(G) = P2 - PI, where P p 1(G) = c and P p2(G) = 1 - c. When Gn is simple majority on n players the central 12 It is crucial to fix s and let n t end to infinity. If s t ends to zero as n tends to infinity the sit uat ion can be very different. For example, when s = lin the majority function is most sensit ive to noise.

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limit theorem implies that te(Gn ) = O(l/J1i). It is known that for nplayer games (n being an odd integer) and every e, te(G) is minimized by the simple majority game 13 . The next theorem asserts that uniform noise stability (in a slightly stronger sense than before) implies that up to a multiplicative constant the aggregation of information is optimal.

Theorem 7.3. Let (Gk) be a sequence of strong simple games such that Gk has nk players and let c > 0 be a fixed real number. Suppose that the class of games and distributions (G k , P p ) , p E Te(Gk) is uniformly noise-stable. Then te(Gn ) =

O(l/yffik).

3. Indeterminacy. Consider a sequence (Gn ) of strong simple games. Let m ~ 3 be a fixed integer , let A be a set of m alternatives, and denote by Fn the neutral social welfare function based on Gn when the set of alternatives is A. Let R be an arbitrary asymmetric relation on A. It was proved in Kalai (2004) that if (Gn ) has diminishing individual ShapleyShubik power then, for n sufficiently large, R is in the range of Fn (in other words, PR(G n ) > 0) and this property is referred to there as complete indeterminacy. Note that social chaos (in view of Theorem 1.7) can be regarded as the ultimate form of social indeterminacy as it implies that all the probabilities PR(G n ) are asymptotically th e same. The main tool for showing that diminishing individual Shapley-Shubik power implies social indeterminacy is the equivalence between diminishing individual Shapley-Shubik power and asymptotically complete aggregation of information. It also follows from the results of Kalai (2004) that diminishing individual Shapley-Shubik power implies diminishing individual Banzhaf power (but not vice versa). When we assume diminishing individual Banzhaf power we can expect the following strong form of complete indeterminacy.

Definition 7.4. A sequence (G n ) of strong simple games leads to stochastically complete social indeterminacy if for every asymmetric relation R on a set A of m alternatives

13 The fact that the derivative dPp(G)/dp is maximized for simple majority can be found , e.g., in Friedgut and Kalai (1996).

Noise Sensitivity and Chaos in Social Choice Theory

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Theorem 7.5. (1) A sequence (On) of strong simple games with diminishing Banzhaf individual power leads to stochastically complete social indeterminacy. (2) A sequence (On) of strong simple games that is uniformly chaotic leads to stochastically complete social indeterminacy. The proof of Theorem 7.5 uses the argument used in proving Theorem 1.7 and for part (1) we require also a recent theorem of Mossel, O'Donnell and Oleszkiewicz (2005), referred to as "it ain't over until it's over". This theorem roughly asserts that for any random voter profile when a large random set of votes is counted there is still (almost surely as n tends to infinity) a probability bounded away from zero (however tiny) that either candidate will win. We defer further details to the Appendix. 4. Noise stability. This paper is mainly devoted to the chaotic behavior of noise-sensitive classes of simple games. There are interesting issues concerning the behavior of social preferences for social welfare functions based on uniformly noise-stable classes of simple games, and random uniform voter profiles. I will mention one topic worth further study: suppose that we are interested in the social preference relation between two alternatives A and B but we can cannot compare these two alternatives directly. Rather, we can compare both A and B with other alternatives G1 , G2 , · · . Gr. Suppose that for all those alternatives we discover that the society prefers A to C, and C, to B. It seems true that as r grows we can conclude that the society prefers A to B with higher and higher probability that tends to 1 as r tends to infinity. This is unknown even for simple majority. (In contrast, it follows from Theorem 1.7 that for every fixed r and a noise-sensitive sequence (On) of strong simple games, the a posteriori probability that the society prefers A to B tends to 1/2.) 5. Other economic models. We study noise sensitivity and its chaotic consequences for social welfare functions. It would be interesting to study similar questions for other economic models and , in particular, for exchange economies and more general forms of economies. Related notions of indeterminacy, aggregation of information, pivotal agents, and power were considered for various models of economies and some analogies with results in social choice theory can be drawn. The well-known Sonnenschein-DebreuMantel theorem concerning demand functions for exchange economies is an indeterminacy result that implies that various dynamics for reaching equilibrium points can be chaotic. Yet, it appears that exchange economies

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G. Kalai

behave in a similar way to weighted majority functions and are thus quite stable to noise in terms of initial endowments and individual demand functions . More complicated models of economies (or simple models under a complicated probabilistic environment) appear to exhibit a more chaotic behavior in the sense of this paper. 6. The asymptotic nature of our results. We would like to stress that our approach and results are asymptotic: we consider the situation when the number of individuals tends to infinity. The number of alternatives m and the noise c are supposed to be fixed as n tends to infinity, so one has to be careful in drawing conclusions for numerical values of n, e, and m. Moreover, the dichotomy between stochastic stability and complete chaos is based on first fixing c and letting n tend to infinity and then letting c tend to zero. Studying the dependence on c even for multi-level majority-based rules gives a more involved picture that we briefly discussed in Section 3.2.

8.

CONCLUSION

Social welfare functions form a simple and basic model in economic theory and political science. We have considered their probabilistic behavior under uniformly distributed voter profiles. In contrast to social welfare functions based on simple majority (or weighted majority) which demonstrate stable behavior to small stochastic perturbations, in this paper we studied a class of social welfare functions that demonstrate completely chaotic behavior and showed that this class can be characterized in surprisingly different ways. This class contains some rather natural examples. Chaotic behavior occurs in our model for hierarchical or recursive aggregation of preferences and in cases where the social preferences heavily depend on a small number of small communities with decisive views. One interpretation of our results relates to the primary role of the majority rule. There have been many efforts to demonstrate the dominance of majority among voting rules. Analysis of sensitivity to noise gives a clear advantage to the majority rule. Our results on the stochastic robustness of majority and the chaotic nature of methods that are asymptotically orthogonal to majority point in this direction. So is the result by Mossel, O'Donnell and Oleszkiewicz (2005) on the asymptotic optimality of the majority rule in terms of sensitivity to noise. The majority rule is a very simple and very basic economic/political mechanism and an analysis of

Noise Sensitivity and Chaos in Social Choice Theory

207

sensitivity to noise in a similar manner to ours may be an important aspect in evaluating other economic mechanisms. Do we witness chaotic behavior in realistic economic situations of preference aggregation? A complete social chaos of the kind considered in this paper is an ideal rather than a realistic economic phenomenon. However, I would expect to find chaotic components in real-life examples of preference aggregation and in my opinion complete (stochastic) stability of the kind we observe for weighted majority is often unrealistic. Understanding the chaotic components (in the sense of this paper) in other stochastic economic models is worthy of further study.

9.

ApPENDIX

9.1. Proofs of properties of simple majority

Proof of Theorem 1.6. Let hk+l be the majority function on the set V = {I, 2, . . . , 2k + I} . In order to prove Lemma 3.1 we will study the FourierWalsh coefficients of hk+ 1. For a Boolean function f the coefficient 1( {j} ) by definition equals L::scv, jrl.s 2- n f(8) - L::scv, jES 2- n f(8). Therefore , if f is monotone we obtain that

1( {j})

= -bj(f)/2 ,

where bj(f) is the Banzhaf value of j in

1(

For hk+l we obtain that {j}) = function f and for k 2: 0 we denoted

f.

k) ek

/2 2k+l . Recall that for a Boolean

For every Boolean function that represents a strong simple game: (1) Wo(f) = ]2(0) = 1/4,

(2)L::~=o Wf(f) = Ilfll~ = 1/2, and (3) Wk(f) = 0 for an even integer k > 0 (Proposition 2.1).

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G. Kalai

For simple majority on 2k + 1 voters we obtain that

It is easy to verify that this expression is monotone decreasing with k. Therefore, if k ~ 1 the value of WI (f2k+l) is maximized for k = 1, namely, for simple majority on three voters . For k = 1 we get WI (13) = 3/16. Of course, h(S) = 0 for lSI> 3 and therefore Wk(f3) = 0 for every k > 3. It follows that for every k ~ 1, WI(f2k+d ~ W1(f3) and Wr(f2k+d ~ Wr(f3). These relations with Theorems 2.3 and 2.2 imply our result. Indeed, Theorem 2.3 asserts that the probability Nc(f) that an s-noise will change the outcome of an election is given by the formula: n

Nc(f)

= 1/2 - 4 ·

L (1 - c:fWr(f). r=l

When we compare this expression for 13 and hk+l the result follows from the fact that when we pass from 13 to hk+l the weights Wi'S (whose sum is constant) are shifted from the lower values of i to the larger ones. Formally, the proof is completed by the following standard lemma :

Lemma 9.1. Let al > a2 > .. ., an > 0 be a sequence of real numbers. Let b = (bl ,b 2, .. . , bn ) and b' = (b~,b~ , ... , b~ ) be sequences of positive real functions such that L:~=l b, = L:~=l b~ and for every r, 1 ~ r ~ n, L:r=l bi ~ L:r=l b~ then L:~=l aibi ~ L:~l aib~. The same argument or just applying Relation 2.7 implies part (i).

9.2. Proof of Theorem 4.3 Theorem 4.3 is the only result in this paper that requires a non-elementary tool from harmonic analysis . We will give the proof for the case where Pk = 1/2, namely, for the uniform probability distribution. In this case the bias comes from the voting rule and not from the probability distribution. Corollary 2.4 (which is quite elementary) extends to the case of Boolean functions that may have large bias, as follows:

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Noise Sensitivity and Chaos in Social Choice Theory

Theorem 9.2 . A sequence (fn) of Boolean functions is noise-sensitive if and only if, for every k > 0, (9.1)

2:i21 Wi(fn) =

qn(1 - qn) , where qn = IIInll~ is simply the probability that In equals 1. (In the case that (fn) represent a strong simple game or represent games with small bias the denominator on the right-hand side of relation (9.1) is bounded from zero and can be deleted .) In view of formula (9.1) relating the noise sensitivity to the Fourier coefficients, in order to demonstrate noise sensitivity we have to show that most of the Fourier coefficients of In (in terms of the 2-norm of In) are concentrated on large "frequencies". For a real function I : On -+ JR, I = 2: f( S)Us, define Note that by the Parseval formula

The Bonamie-Beckner inequality (see, e.g., Benjamini, Kalai and Schramm (1999)) asserts that for every real function I on O(n),

For a Boolean function I, note that for every t, Ilfll~ = P(f) . It follows from the Bonamie-Beckner inequality (say, for e = 1/2) that for some constant K,

L o
P(S) S K J(qn (1 -

qn)) .

This is sufficient to show that relation 9.1 holds even when k depends on n and is set to be (1/4) . log (1/( qn' (1- qn))).

9.3. It ain't over until it's over Sketch of the proof of Theorem 7.5. In the proof of Theorem 1.7 we considered two asymmetric relations Rand R' on m alternatives that differ only in the ranking of the two last alternatives and used noise sensitivity to show that the ratio PR(Gn)/PR,(Gn) tends to one as n tends to infinity.

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G. Kalai

If we want to show only that the ratio PR(Gn)/PR,(Gn) is bounded away from 0 we need a weaker property for the sequence (Gn ) , referred to as [IAOUIOj, which is defined as follows:

Let G be a monotone simple strong game considered as a voting rule between Alice and Bob with n voters. Let e > 0 and 8 > 0 be small real numbers. Given a set S of voters and the way these voters vote, let Q denote the probability when the remaining voters vote uniformly at random that Alice will win the elections. Let q(G; e, 8) be the probability that a random choice of a set S of (1 - c)n of the voters and a random choice of the way voters in S voted the value of Q is at least 8. Consider a sequence (Gn ) of strong simple games regarded as voting rules. The sequence (Gn ) has property [IAOUIOj if • For every c

> 0 there is 8 > 0 so that lim q(G; c, 8) = 1.

n-+oo

It can be shown using the central limit theorem that the sequence of simple majority games on n players has the property [IAOUIOj. In this case 8 can be taken to be «<" , As indicated above the proof of Theorem 1.7 gives Theorem 9.3. A sequence (Gn ) of strong simple games with property [IAOUIOj leads to stochastically complete social indeterminacy. Showing property [IAOUIOj for a uniformly chaotic sequence of games is immediate from the definition and therefore part (2) of Theorem 7.5 follows directly from the proof of Theorem 1.7. To prove the first part we need to show that diminishing Banzhaf individual power implies the property [IAOUIOj . This is much more difficult and was recently proved by Mossel, O'Donnell and Oleszkiewicz (2005) in response to a conjecture opsed by the author and Friedgut.

Theorem 9.4 (It ain't over until it's over [Mossel, O'Donnell and Oleszkiewicz]). A sequence of strong monotone simple games (Gn ) with diminishing individual Banzhaf power has property [IAOUIOj.

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REFERENCES

[1]

K. Arrow, A difficulty in the theory of social welfare, J. of Political Economy , 58 (1950), 328-346.

[2]

C. E. Bell, A random voting graph almost surely has a Hamiltonian cycle when the number of alternatives is large, Econometrica, 49 (1981), 1597-1603.

[3]

1. Benjamini, G. Kalai and O. Schramm, Noise sensitivity of Boolean functions and applications to percolation, Pub!. 1.H.E.S. (1999).

[4]

T . Feddersen, and W. Pesendorfer, The swing voter's curse , The American Economic Review, 86 (1996), 408-424 .

[5]

T . Feddersen, and W . Pesendorfer, Convicting the innocent : the inferiority of unanimous jury verdicts under strategic voting, The American Political Science Review, 92 (1998), 23-35 .

[6]

P. C. Fishburn, W. V. Gehrlein and E. Maskin , Condorcet proportions and Kelly's conjectures, Discrete Appl. Math., 1 (1979), 229-252.

[7] T . E. Friedgut, Political Parti cipation in the USSR, Princeton University Press, Princeton, N.J . (1979). [8] E. Friedgut, and G. Kalai , Every monotone graph prop erty has a sharp threshold , Proc. American Mathematical Society, 124 (1996), 2993-3002. [9]

W. V. Gehrlein, Condor cet 's paradox and the Condorcet efficiency of voting rules, Math. Japon., 45 (1997), 173-199.

[10] R. Holzman , E. Lehrer and N. Linial , Some bounds for th e Banzhaf index and other semivalues, Math. Oper. Res., 13 (1988), 358-363 . [11]

Kalai , G., A Fourier-theoretic perspective for the Condorcet paradox and Arrow's theorem , Adv. in Appl. Math., 29 (2002),412-426

[12]

G. Kalai , Social indeterminacy, Econometrica, 72 (2004), 1565-1581.

[13]

S. Khot , G. Kindler, E. Mossel and R. O'Donn ell, Optimal inapproximability results for MAX-CUT and ot her 2-variable CSPs ? SIAM Journal of Computing, 37(1) (2007), pp . 319-357 . (Also FOCS 2004).

[14]

E. Mossel and R. O'Donn ell, On the noise sensitivity of monotone functions , Random Stru ctures Algorithms, 23 (2003), 333-350 .

[15]

E. Mossel, R. O'Donnell and K. Oleszkiewicz, Noise stability of functions with low influence: invariance and optimality, Annals of Mathemati cs, to appear 2010. (Also FOCS 2005.)

[16]

Y. Peres , Noise stability of weighted majority, preprint math.PR/0412377 (2004).

[17]

D. G. Saari , Chaotic elections! A mathematician looks at voting, American Mathematical Society, Providence, R 1. (2001).

[18] Y. Samet, Equilibria with information aggregation in sharp threshold voting rules, M.Sc. th esis, Jerusal em, 2004.

G.K~ill

212

[19] W. Sheppard, On the application of the theory of error to cases of normal distribution and normal correlation, Phil. Trans. Royal Soc. London, 192 (1899), 101-168.

Gil Kalai Hebrew University of Jeru salem and Yale University e-mail:

kalai~ath.huji.ac .il

Fete of Combinatorics, pp . 213-238.

SOlYAI SOCIETY MATHEMATICAL STUDIES, 20

COLORING UNIFORM HYPERGRAPHS WITH SMALL EDGE DEGREES

ALEXANDR V. KOSTOCHKA*, MOHIT KUMBHAT and VOJTECH RODLt Dedicated to the 60th birthday of Laszlo Loudsz

Let k be a positive integer and n = Llog2k J. We prove that there is an e = c(k) > that for sufficiently large r , every r-uniform hypergraph with maximum edge degree at most

o such

c(k)

e (-.!.-) n+l ln r

is k-colorable .

1.

INTRODUCTION

The degree of an edge e in a hypergraph G is the number of other edges of G intersecting e, and the maximum edge degree of G is the maximum over the degrees of its edges. A natural question is: Which bound on the maximum edge degree of an r-uniform hypergraph G provides that G is k-colorable? The classical result in this direction belongs to Erdos and Lovasz. In their seminal paper [2] (where the Lovasz Local Lemma appeared), they proved the following bound . •Resear ch of this author is supported in part by NSF grant DMS-0650784 and by grant 06-01-00694 of the Russian Foundation for Basic Research . tResearch of this author was supported by NSF grants DMS-0300529 and DMS0800070 .

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A. Kostochka, M. Kumbhat and V. Rodl

Theorem 1 [2]. If k , r ~ 2, then every r-uniform hypergraph with maximum edge degree at most ikT is k-colorable.

The proof works also for list coloring. A remarkable feature of this result is that it works for all k, r ~ 2, and in many cases the bound is rather close to the best possible. In particular, Erdos and Lovasz [2] showed that the bound cannot be significantly improved even if we consider only hypergraphs with high girth. If we denote by D(k, r) (respectively, D(k, r, g)) the minimum D such that there exists an r-uniform non-k-colorable hypergraph G with maximum degree D (and girth at least g), then the results mentioned above can be summarized as follows. For every 9 ~ 2, 3k ~e 4 < D(k , r) < - 20r

(1)

T

-

1

.

The upper bound in (1) was recently improved for k < r by Kostochka and Rodl [5] to r kT-lln k1-

rr

Let m(r, k) denote the minimum number of edges in an r-uniform hypergraph that is not k-colorable. Elaborating the proof of a lower bound on m(r, 2), and using the Local Lemma, Radhakrishnan and Srinivasan [6] improved the lower bound on D(2 , r) for large r to (2)

D(2, r) 20.17 · 2T Jr/ In r,

The main result of this paper is the following extension of (2) to fixed k and large r, Theorem 2. For every integer k ~ 2, let E = E(k) = exp {-4k 2 } and n = n( k) = [log., k J. Then for every sufHciently large r, every r-uniform n hypergraph with maximum edge degree at most D = EkT(I;T) n + l is kn

colorable. In other words, D (k, r)

> EkT( I;T) n+ 1 •

Recall that for the class of simple hypergraphs, Kostochka and Kumbhat [4] recently proved a slightly stronger bound than Theorem 2 gives. Recall that a hypergraph G is b-simple if [e n e'l :::; b for every distinct e, e' E E(G). Theorem 3 [4]. If b ~ 1, k ~ 2, and E > 0 are fixed and r is sufHciently large, then every r-uniform b-simple hypergraph G with maximum edge degree at most kTr1- c is k-colorable.

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Coloring Uniform Hypergraphs with Small Edge Degrees

We think that the statement of Theorem 3 holds for all r-uniform hypergraphs. The proof of Theorem 2 uses the Local Lemma, ideas of Radhakrishnan and Srinivasan [6], and the proof of the lower bound on m(r, k) by Kostochka [3] . The structure of the paper is as follows. In the next section, a semirandom procedure Evolution is described and some of its simple properties are derived. In Section 3 we study the structure of so called cause trees arising in the analysis of Evolution. In the next two sections we define some auxiliary "bad" events and estimate their probabilities. Using the independence structure of these auxiliary events and the Local Lemma, in the final section we show that for hypergraphs satisfying the conditions of Theorem 2, with positive probability Evolution gives a proper k-coloring. This means that such a coloring exists.

2. COLORING PROCEDURE EVOLUTION AND ITS PROPERTIES

Let k, n and

E

be as in the statement of the theorem. Let

(3) Throughout the paper we will use the notation

(4) Fix some 0 < p < 2- k rr . Then there is the unique positive integer s such that sp :S (;~~)r < (s + l)p . Let G = (V, E) be an r-uniform hypergraph n

with maximum edge degree at most D = Ekr (l~ r ) n+l • The coloring procedure Evolution described below consists of n + 1 stages, and every stage apart from Stage 0 consists of s steps . For 1 :S l :S n and 1 :S i :S s, Step (l - l)s + i is the ith step in Stage l. We also fix a linear order L on V(G). Now, the procedure works as follows. Stage O. (01) Color every vertex v E V(G) randomly and independently, with a color <jJ (v) E {a, 1, 2, .. . , k -I} chosen uniformly in this set.

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(02) For every v E V(G), define the random variable I(v) with the values in {1, 2, ... , sn} U {oo} as follows:

Pr { I(v) = x } =

(5)

p,

if x E {1,2, .. . ,sn};

{ 1 - psn, if x = 00.

Each random variable I(v) is defined to be mutually independent of all other

I(w). Stage 1, 1 = 1, ... .n: STEP i + s(l - 1), 1 ::; i ::; s. Following order L, for one by one vertex v E V(G), check whether

(C1) I(v) = (1- l)s + i and (C2) v belongs to an edge that was monochromatic, say, of color a, before Stage 1, and still is monochromatic at the current moment. If both conditions (C1) and (C2) hold, then recolor v with color a+ 21(modulo k). Otherwise, do nothing with v.

1

Remark 1. By Condition (C1), each vertex can be recolored at most once. Remark 2. As it follows from the description of the procedure, every step consists of V(G)I smaller steps (one per vertex).

I

Lemma 1. For every w, q 2: 1, every set W ~ V with IWI = w, and every set Q ~ {1,2, ... ,sn} with IQI = q, the probability that for each vertex v E W, I(v) E Q is at most (qp)w. Proof. For every vertex v E V (G) and every 1 ::; 1 ::; nand 1 ::; i ::; s, Pr {I(v) = s(l - 1) + i} = p. Therefore, the probability that I(v) E Q is at most qp. The mutual independence of all I(v) yields the lemma. _

For an edge e E E and 1 ::; 1::; n , let

(6)

M(e,l) = { vEe: I(v) ::; sl}.

Lemma 2. For every e E E and 1 ::; 1::; n, Pr { IM (e, 1) I 2: z} ::;

5r• EO.

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Coloring Uniform Hypergraphs with Small Edge Degrees

I

Proof. It is enough to prove the lemma for l = n. Observe that M(e, n)1 ~ z means that there exists a set Z C e with IZ! = z such that I(v) ::; sn for every v E Z. Hence by Lemma 1 applied with Q = [sn] and W = Z , this probability is at most

r) z (er) z ( z (nsp) ::; ~

(

n In r ) (n + l)r

z

::;

(ne In r ) z(n + 1)

z

Since r is large and z = l4k 2r / In r J > n, 2

eln r e In r ne In r ------,<-<< T - 0 .6 2

z(n + 1) - z + 1 -

4k r -

Thus

Lemma 3. If a vertex is of color Q at tbe end of Stage l, l ~ 1, tben at tbe end of Stage 0 it can be colored only witb colors Q, Q - 20 , Q - 21 , ... , 1 1 (modulo k). Q - 2 Proof. By Remark 1, every vertex can be recolored at most once and by definition , a vertex of color f3 can be recolored during Stage j only with color f3 + 2j - 1 (modulo k). • . Definition [Blaming edges]. If an edge eo becomes monochromatic of color Q during Stage l, then it must contain at the end of Stage 0 a vertex of color 1 1 Q - 2 - . Suppose that at the end of Stage 0 it contained vertices of colors 1 Ih 1 Q - 2h - , ... , Q - 2 - , where Ih = I and h < l2 < .. . < Ih. Then for every 1 ::; j ::; h, there exists an edge ej and a vertex Vj E eo n ej such that (a) ej was monochromatic of color Q - 2l j - l at the end of Stage Ij - 1; (b) Vj was recolored with Q during Stage Ij and it was the last vertex of this color in eo recolored with Q. In this case we say that eo and Vj Ij-blame ej' Remark 3. If an edge eo becomes monochromatic of color Q during Stage I, then all vertices of color Q - 21- 1 in eo change their colors to Q during this stage. But since in every step , the vertices of G are considered consecutively, there is the last vertex v* of color Q - 21- 1 in eo that changes its color. At that moment, any monochromatic edge e of color Q - 21- 1 containing v* shares only v* with eo.

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Remark 4. It may happen that an edge eo can blame more than one edge containing the same vertex Vj' On the other hand, by definition, eo cannot blame an edge containing another vertex v E eo with ¢>(v) = ¢>(Vj) .

Definition [Cause trees]. If an edge eo is monochromatic of color a at the end of Stage l, then a cause tree T = T(eo, a, l) is a subset of edges of G defined by induction on l as follows: (a) The set T always contains eo. (b) If eo was monochromatic of color a already after Stage 0, then T = {eo} for every l . (c) Suppose that at the end of Stage 0 edge eo contained vertices of colors a-2 h -1 , .. . ,a_2Ih - l , where lh ::; l and it < h < ... < lh. Suppose further that for j = 1, ... , h, edge eo lrblames edge ej' Then h

T = T(eo, a , l) = {eo} U

UT(ej, a - 2 -1, lj -1). 1j

j=l

Remark 5. By Remark 4 and the definition of cause trees , it could be that in the same outcome of Evolution for the same triple (eo, a , l) , we can construct several distinct cause trees T = T(eo, a , l).

Definition [Levels of edges]. If T = T( eo, a , l) is defined as above, then we also say that el , e2 , ... .ei; are the edges of level 1 of T, the edges blamed by the edges of level 1 are the edges of level 2 of T , and so on. Thus, if an edge e of a cause tree has vertices of exactly t distinct colors at the end of Stage 0, then e blames either t - 1 or t other edges.

3.

STRUCTURE OF CAUSE TREES

Since each vertex can be recolored at most once, each edge at different stages of Evolution can become monochromatic with at most two colors. Furthermore, if an edge e was monochromatic of a color al after Stage it and becomes monochromatic of a color a2 i= al after Stage la, then e has to be monochromatic of color al already after Stage 0 and all vertices of e change their color to a2 = al + 212 - 1 at Stage la- In this case, each cause tree for e considered after Stage h has exactly one edge of level 1.

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Coloring Uniform Hypergraphs with Small Edge Degrees

In view of this, if an edge e becomes monochromatic exactly once during Evolution, then the corresponding color a is called the main color of e and denoted by 1-£( e), and if e becomes monochromatic twice, then the main color of e, 1-£( e), is the first of these two colors. If e is monochromatic of some color a after some Stage I, then we say that e is an I-unlucky edge. Lemma 4. If eo is an I-unlucky edge with a cause tree T, then the main colors of all the edges of T are distinct. Proof. If e and e' are edges of T, then there exist two sequences eo,e1 , . . . , eq = e and eo = eo,eL . .. , e~, = e' such that ej lj-blames ej+1 for j = 0,1 , ... , q - 1 and ej lj-blames ej+1 for j = 0,1, .. . , q' - 1. Furthermore, lo > h > ... > Iq-1, lo > l~ > ... > 1~_1' and the sequences lo , h, , lq and lo, l~ , . . . ,l~ are not identical. Thus , the numbers 210- 1+ 211- 1+ +21q-1- 1 and 21~-1+21~-1+ .. . +21~'_1-1 are distinct and differ by less than k . On the other hand , by definition , the main color of e is a_210-1_211-1_ . . ·_21q-1-1 and the main color of e' is a - 21~ -1 lemma. •

-

21~ -1 -

... -

21~, -1 -1. This proves the

Lemma 5. Suppose that eo is an I-unlucky edge with a cause tree T . If e and e' are edges of T and neither of them blames the other, then e and e' are disjoint. Proof. Assume that e and e' have a common vertex v and both belong to T. Then there exist two sequences eo, e l, "" eq = e and e~ = eo, e~ , .. . , e~, = e' such that ej lj-blames ej+1 for j = 0,1 , . . . , q - 1 and ej lj-blames ej+1 for j = 0,1 , .. . , q' - 1. Furthermore, 1o > h > ... > 1q-1, l~ > l~ > ... > 1~_1' Claim 1. Iq-1

=1=

l~, -1 .

Proof of Claim. If Iq-1 = l~, -1' then e and e' both were monochromatic at the end of Stage 1q-1 -1. But by Lemma 4, their main colors differ. This proves the claim. Thus below we can assume that Iq-1 < l~, -1' It follows that e ceased to be monochromatic before e' did. In particular, the vertex vEe n e' was recolored from 1-£( e) to 1-£( e'). This yields that

(7)

I-£(e') - I-£(e) (modulo k) is a power of 2.

Claim 2. I-£(e/) - I-£(e) = 21q-1-1 modulo k.

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Proof of Claim. Recall that

J-l(e') - J-l(e) = (a - 210-1 - 21~-1 - .. . - 21~I_l-1)

_ (a -

210 -

1-

2!I- 1

- .. . _

21q- 1 - 1) .

In this expression, a cancels out and every other summand apart from 21q- l - 1 is divisible by 21q - l . Together with (7), this yields the claim. Claim 2 implies that v was recolored during Stage lq-1 and thus J-l(e') = J-l(e q-1). This contradicts Lemma 4. _

Lemma 6. Let eo be an edge of G that is unlucky after Stage 1. Let T be a cause tree for this event such that for every pair of edges (e1' e2) of T such that e1 blames e2, we have J-l(e1) - J-l(e2) E {I, 2, ... ,2 1-1} (modulo k). Then E(T) 21. In particular, each cause tree has at most 2n :::; kedges.

I

I : :;

Proof. If e1 lr-blames e2 and e2 h-blames e3 , then 12 < lr. Thus, under conditions of the lemma , for the root eo and an arbitrary edge e of the tree, we have J-l (eo) - J-l (e) E {I, 2, ... , 21-1 + 21-2 + ... + I} = {I, 2, . .. , 21 - I}. Now, Lemma 4 implies that T has at most 1 + (21 - 1) edges. _ Below we will analyze which subsets of edges of G can form cause trees T(e, a, 1) for some values of e, a and l , Lemma 5 implies that every cause tree T = T( e, a, 1) is an r-uniform hypergraph tree in the ordinary sense rooted at e. Moreover, every vertex of such a tree belongs to at most two edges of this tree. By Lemma 6, such a tree has at most k edges. In connection with this, let us fix some notation.

Definition [r-Trees and sub-r-trees]. When we say ''r-tree'', we mean an runiform hypergraph tree with at most k edges in which every vertex belongs to at most two edges of this tree. By a sub-r-tree of G we mean an r-tree that is a subhypergraph of G. Often , we will consider rooted r-trees. The root of an r-tree will be an edge of this r-tree, and not a vertex. Given an r-tree T with a root eo, the children of eo are the edges adjacent to eo, and for e E E(T) at distance d from eo (in T) , the children of e are the edges adjacent to e that are at distance d + 1 from eo. Naturally, the descendants of an e E E(T) are its children , children of children and so on. If e1 is a descendant of e2, then e2 is an ancestor of e1 . For an r-tree T with a root eo and another edge e1 ofT, by T(e1) we denote the subtree of T formed by e1 and all its descendants. We will use the following fact on sub-r-trees of r-uniform hypergraphs.

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Coloring Uniform Hypergraphs with Small Edge Degrees

Lemma 7. Let H be an r-uniform hypergraph with maximum edge degree at most D. Let eo E E(H) . Then eo belongs to at most (4D)y-1 sub-r-trees of H with y edges.

I

I

Proof. Let T be a sub-r-tree of H containing eo with E(T) = y. Consider T as a rooted r-tree with root eo. Order the edges of Teo, el," ., ey-l starting from eo using Breadth-First search . We say that T has type (ho, , hy- 2 ) if for i = 0, ... ,y - 2, edge ei has exactly hi children. Since ho + + h y- 2 = Y - 1, the number of distinct types does not exceed the number of representation of y - 1 as the sum of y - 1 of ordered nonnegative summands, which equals (Y-I)t~2-1)-I) < 4y - l . When we know the type of T, then for every edge ei, i ~ 1, we know the immediate ancestor (father edge). So, we can embed a tree T of a given type, edge by edge into G. Furthermore, at each step i, i ~ 1, we have at most D choices for our edge among the edges of G adjacent to its father edge. Thus, eo belongs to at most Dy-l r-trees of given type with y edges. Since the number of distinct types is at most 4y - l , this proves the lemma. _

4.

AUXILIARY EVENTS

The goal of this section is to introduce the auxiliary events W(eo, a, T , l) that imply the "bad" events in Evolution and can be controlled. In the next section we estimate probabilities of these auxiliary events .

Definition [Events W(e,a,T,l)] . Let e E E(G), a E [k], l E {l, ... ,n} and T be a sub-r-tree of G rooted at e. Then let W (e , a, T , l) be the event that edge e is monochromatic of color a after Stage l of Evolution, and a cause tree for this is T .

Lemma 8. Let e E E(G) , a E [k], l E {I , ... ,n } and T be a sub-r-ttee of G rooted at e. Also, let el, ... , eq be the edges of T of the first level, i.e., the edges ofT sharing a vertex with e. For j E [q], let en ej = {Vj}. Let Q ~ {VI, . . . , v q } . If W (e, a, T, l) occurs, then the following properties hold: (WI) For every vEe, (v) E {a} U (Q) ~ {o;o - 2°, a - 21, .. . , a - 21(modulo k).

1}

(W2) For j E [q], (Vj) =I a , and for distinct j and j', (vj') =I (Vj). In particular, if Aj = Aj (e, ( v) = (Vj)}, then all sets Aj are disjoint subsets of e.

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(W3) I(v) :::; Is for each v E UJ=l A j . Moreover, for each j E [q], if Vj becomes of color a at Stage Ij, then (W4) a -
\

(W5) the event W(ej, (Ij -1)s, we have also I(u) ;::: I(vj); and (W7) for each u E A j - Vj, (Ij -1)s + 1:::; I(u) :::; I(vj). (W8) If e was already monochromatic after Stage I-I, then for each vEe, I(v) ~ [s(1 -1) + 1, sl] .

Proof. We will prove (Wl)-(W8) one by one. (WI) holds by the definition of cause trees and Lemma 3. By the definition of cause trees, for each j E [q], Vj is the last vertex of color
Now (W7) follows from the facts that all vertices in Aj must change their colors in Stage Ij (in order to change it from
Unfortunately, events W(e, a, T, l) and W(e', a' , T ', l') can be dependent even if V(T') is disjoint from V(T). For example, some edge eo E E(G) may intersect both V(T) and V(T'). In this case, event W(e, a, T ,I) may prompt a vertex v E eo n V(T) to change its color. This in turn, can make

223

Coloring Uniform Hypergraphs with Small Edge Degrees

eo monochromatic and so prompt a vertex v' E eo n V(T') to change its

color, which may affect W(e', 0/, T', 1'). Therefore, for each eo E E(G), each sub-r-tree T of G with root eo and E(T) k, and each color a, we will introduce the auxiliary event

I

I ::;

W(eo,a,T,l) that contains the event W(eo,a,T,l), and in addition essentially possesses properties (Wl)-(W8) above, but does not depend on the values of ¢>(u) and I(u) for all u ~ V(T). This will imply that each W(e, a, T, 1) is independent of all W(e~, a' ,T' , 1') with V(T') n V(T) = 0. We define these events by induction on the number of edges in T. If E(T) = {eo}, then the event W(eo,a,T,l) means that all of the following holds (i) ¢>(eo) is monochromatic of color a,

(ii) IM(eo, n)1 < z , and (iii) I(v) > Is for every v E eo . Suppose that the event W(eo, a, T, 1) is defined for all parameters eo, a, T , 1such that IE(T)I < y. Let eo E E(G) , a E [k], and T be any sub-r-tree T of G with root eo and y edges. Let el , .. . ,eq be the edges of T sharing a vertex with e. For j E [q], let e n ej = {Vj}. Let Q ~ {VI , ... , vq } . We say

1M

that W (eo, a ,T , 1) occurs, if either (e, n) I ~ z for at least one e E E(T) or all of the following holds: (WI) For every vE e, ¢>(v) E {a}U¢>(Q) ~ {a, a-2°, a-2 1 , . . . , a-21- 1 } (modulo k).

(W2) For j E [q], cP(Vj) i= a, and for distinct j and i'. cP( vj' ) i= cP(Vj). In particular, if Aj = Aj (e, ¢» = {vEe : cP(V) = cP( Vj)} , then all sets A j are disjoint. (W3) I(v) ::; ls for each v E U]=I Aj. Moreover, for each j E [q], if (lj - l)s + 1::; I(vj) ::; slj , then (W4) a -¢>(Vj) = 21r

l;

(W5) event W(ej, ¢>(Vj), T(ej), Ij - 1) occurs; (W6) for every u

E ej

with I(u) > (lj -1)s, we have also I(u)

~

I(vj), and

(W7) for each u E Aj - Vj , (lj - l)s + 1 ::; I(u) ::; I(vj). (W8) If event W(eo, a, T, 1 - 1) occurs, then for each v E eo, I(v) ~ [s(l - 1) + 1, sl] . The following two lemmas justify the introduction of the events Wee, a, T , l).

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Lemma 9. Let eo E E( G), a E [k], l E {O, . .. , n} and T be a sub-r-tree ofG with root eo. If the event W (eo, a, T, l) occurs, then the event W (eo, a, T, l) also occurs.

Proof. Suppose that forsome values of the parameters eo, T, l, and a, W(eo, a, T, l) occurs but W(eo , a, T, l) does not occur. We may assume that l is minimal with this property, i.e., that for all quadruples (e~, a' , T ', l') with l' < l the lemma holds. Let us check which of the properties in the definition of W(eo, a, T, l) may fail. Since (WI) and (W2) coincide with (WI) and (W2), respectively, they hold. For the same reason, properties (W4), (W6), and (W7) hold. Property (W5) follows from (W5) and the minimality of our counterexample. Property (W3) follows from the fact that otherwise, by the definition of Evolution (Condition (Cl)) , some vertex in U]=l A j would not change its color to a .

-

-

Assume finally that (W8) does not hold, in other words, that W (eo, a, T, l - 1) occurs, and for some v E eo, I(v) E [s(l - 1) + 1, sl] . By (W8), this implies that W(eo, a, T, l - 1) does not occur, i.e., after Stage l - 1, eo is not monochromatic of color a . It follows that in order eo to become monochromatic of color a after Stage l, we need I(u) E [s(l-l) + 1, sl] for some u E Uj=l Aj. On the other hand , by (W3) for the event W(eo, a, T, l- 1), I(u) ::; (l-l)s for each u E Uj=l A j . This contradiction finishes the proof of the lemma. _ Lemma 10. Let eo E E(G), ao E [k], lo E {O, ... , n}, and To be a sub-rtree of G with root eo. Then W(eo, ao, To, lo) is independent of all events W(e, a, T, l) such that V(T) n V(To) = 0.

Proof. By definition, the events W ( eo, ao, To, lo, '!f'( eo)) are completely defined when we know the values of ¢(v) and I(v) for all v E V(To). This yields the lemma. _

5. PROBABILITIES OF THE AUXILIARY EVENTS The whole proof is based on the following lemma.

225

Coloring Uniform Hypergraphs with Small Edge Degrees n

Lemma 11. Let D := EkrC~r) n+l and G be an r-uniform hypergraph with maximum edge degree at most D . Let e E E(G), a E [k], and 0 :::; l :::; n . Let T be a rooted sub-r-tree of G with root e. If T has y edges, then _ Pr (W(e,a ,T,l)) :::;

( r ) n+~ lnr n

ED-Y

•

Proof. We start from proving the (simple) auxiliary inequality D k EO.5r

(8)

<_

E_.

- lOOk

By definition and (3),

Since E4 = e- 16k2 < _1_ lOOk' (8) would follow from the fact that which is equivalent to (k2r)k:::;

(k2r)kcO .5r c,

<1

-,

(e 4k2 ) o.5r .

The last inequality is obvious. _ To prove the lemma, we

u~

induction on l . Consider first l = O. If

Wee, a, T, 0) occurs, then by (WI), ¢(v) = a for each vEe. Thus, in this case

n

Pr (W(e,a,r,O)) = k:" =

~ (l~r) n+1

•

This proves the case l = O. Now, suppose that the lemma holds for every l' < l. Consider the event W(e ,a,T,l) for some e E E(G), an r-tree T with y edges rooted at e, and a E [k]. Let X(T) denote the event that for at least one e' E E(T), the set M(e',n) = {v E e' : I(v) :::; sn} (cf. (6)) has at least z = l4k2r / ln r J elements. More shortly,

(9)

X(T) =

U

{IM(e',n)1 2:: z}.

e'EE(T)

Let X(T) denote the complement of X(T) . TV (e , Ct, T, l) n X (T) occurs.

Suppose that the event

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When a sub-r-tree T with root e is given , it automatically defines the edges el,' .. , eq of T that share a vertex with e. It also defines for every j E [q], the vertex {Vj} = en ej and the number Yj of edges in T(ej) . Below, when T is given, we will denote Q = Q(T, e) ~ {VI, ... , vq } . Every outcome of Evolution such that W(e, 0:, T, l)nX(T) occurs defines the vector

(h,... ,lq) such that s(lj - 1) < I(vj) ::; slj for each ~

j E [q] . By (W3), for

~

each j E [q], lj E [I] . By (W4) and (W2), alllj are distinct. Let 80

(10)

= 8 0 (q, I) be the set of vectors (h, ... ,lq) such that all

h, ... , lq are

distinct and belong to [I].

By the previous paragraph,

(11)

W(e,o:,T,I) nX(T) = W(e,o:,T,I) nX(T) n {(lI, ... ,lq) E 8 0 } .

Let 8 1 (q, l) = 8 o(q, I-I), i.e. the set of (h, ... , lq) E 80 such that lj ::; I-I for all j E [q] . Let 82 = 8 2(q,l ) = 8 o(q,l ) - 8 1 (q, I). For i = 1,2, let

Fi(e, 0: , T, I) = W(e, 0:, T, I) n X(T) n {(h, ... , lq) E 8 i }

.

By (11),

(12)

W(e, 0:, T, I)

~

X(T) u Fl( e, 0:, T, I) U F2(e , 0: , T, l).

Our goal is to prove that for i (13)

= 1,2 ,

Pr (Fi ( e, 0:, T, I)) ::; O.4ED-Y C~

Since by Lemma 2 and (8) , Pr (X(T)) ::; kEO.5r will imply the lemma.

r)

n- l n+ l •

< O.IED-Y, (12)

and (13)

Observe that the condition "lj ::; 1- 1 for all j E [q]" in the definition of ~

~

~

8 1 (0:, I) implies that if W( e, 0: , T ,I) occurs, then all conditions (Wl)-(W8) are satisfied for the event W(e, 0:, T ,1- 1). Thus,

(14)

F1 (e, O:,T, I)

~

W(e,o:,T,I-I).

By the induction assumption,

(15)

~ ( r ) (n-l+l) /(n+l) Pr (W(e , 0:, T, I-I)) ::; ED-Y In r .

227

Coloring Uniform Hypergraphs with Small Edge Degrees

Let Z (e, l) be t he event that I(v) > sl for each v Ee - M(e, l - 1). If W(e , a , T , l) n X(T) holds, then by (W8) , Z( e,l) occurs. Thus ~

~

(16)

F1(e,a,T,l)

~

Z( e,l) .

Since we already know whether the event W( e, a , T ,l - 1) occurs or not after Step s(l - 1), for each vE e - M (e,l-I) , ~

P r (I(v)

sl IW (e,a ,T ,l -I )} ) ~

Pr ( { I (v»

> sl)

(() ( )) Pr I v > s l- 1 l-lps 1 - (l - l)ps '

Since all random variables I (v ) are mutually independent,

(17)

~

Pr ({ Z(e,l) I W( e,a,T,l-I)}) ~

(

1 -lps )r-1M(e,l-1) 1 1 - (l -l)ps

~ (1- psr-1M(e,I-1)1 .

Therefore, P r ({ Z( e,l) I W( e, a , T ,l - I)} )

L

~

P r { M = M( e,l-I)} (1 - psr- 1M1.

M Ce

By Lemma 2, Pr

(I M (e, l - 1) I ~ z ) ~ EO.5r . Hence

L P r {M = M( e,l-I)} (1- psr- 1M1 Mr;;, e

~

EO .

5r

+

L

Pr {M = M(e ,l -I)} (1- psr- 1M 1

Mr;;,e:\MI
~ EO.5r + (1- psr- z ~ EO .5r + exp { -psr ( 1 - ~; ) } . Since ps ~ (;~r)r exp { - psr

- p, by the definition of p and s ,

(1-::;)}~

exp { -

(~:1-pr)

< exp {_ ~ + 4k

-

n+l

2

n+l

(1- ~;) } +pr}.

228

A. Kostochka, M. Kumbhat and V. Rodl

Recall that 4k 2 = -lnE. Since p < 2- krr, pr

< ~~21 and hence

2 r -1 2/( ) 1-1 In exp { - + -4k- + pr } < rn+1e8k ri-l-l < -rn+1. n+l n+l -E

By (3), In r :2:

(18)

1

2E- 2(n+1 )

n +1 ~ -r ...=.l..

E

and by (4),

-1 r ) n+1 E In r 1

-

(

-

E

= exp {-4k 2 } . So, -1

E2

-1

n+1 exp{-4k 2 } < 0.1 ( - r ) n+1 . = (r) In r

In r

By (14) and (16),

Pr (F1(e, a, T, l)) ~ Pr (Z(e, l) n W(e,a, T, l-I)) . Thus by (15) and (18),

Pr ( F 1 (e, a , T, l) ) ~ Pr ({ Z(e, l) I W(e, a, T, l-I)} ) Pr (W(e, a, T, l-I))

~

(Eo.5r + 0.1 C~r) n+\) Pr (W(e,a, T, l-I)) n -l

~ EO.5r + O.IED-Y C~ r) n+1 . Since by (8), EO.5r -

1

< 0.03D-k

~

0.03D-Y, this implies (13) for i = 1.

Now we will prove (13) for i = 2. Suppose that F2(e, a, T, l) occurs. • Then, by definition, l = (h , ... , lq) E 82 = 8 2 (q, l). In particular, there exists j* E [q] such that lj' = l. • By (W5), for every j E [q], the event W(ej,a - 2Ij-l,T(ej),lj -1) occurs. • Also, for every j E [q], there exists hj E is] such that I(vj) = s(lj - 1) + hj. • By (W6), for every j E [q], for every have also I(u) ;::: (Ij - 1)8 + hj'

u

E

ej with I(u) > (lj - l)s, we

• By (WI) and (W2), coloring ¢(e) belongs to the set '1J(h, ... , lq) of colorings 'lj; of e such that the following two properties hold: (PI) 'lj;(Vj) = a - 2lj - l for all j E [q].

229

Coloring Uniform Hypergraphs with Small Edge Degrees

(P2) 'l/J(v) E { 'l/J(VI) , ... ,'l/J(Vq) , a} for all vEe. For j E [q], let Aj = {v E e : ¢(v) = ¢(Vj)} and aj = IAjl-I. • By (W7), for every j E [q], for each U E Aj -Vj, (lj -l)s+ 1 ~ I(u) ~ (lj - l)s + hj . Thus, in order for F2 ( e, a, T, l) to occur, all of the following events should occur: (R1) There exists a vector l = (h, .. . , lq) E 82 = 8 2(q, l) such that for every j E [q], the event RI(j,lj) := W(ej, a - 2Ir l ,T (ej ),lj -1) occurs. Let RI(l) := E

nY=1 R1(j,lj).

(R2) For this vector l = (h, . . . , lq), there exists a vector h = (hI, .. . , hq) [s]q such that for every j E [q], the event R2(j , lj, hj) := {I(vj) =

s(lj -1)

+ hj}

occurs. Let R2(l, h) :=

nY=1 R2(j, lj, hj).

(R3) For these vectors l = (h, .. . , lq) and h j E [q] and every u E ej, the event

occurs.

Let R3(j,lj,hj) :=

nUEerVj

= (hI, ... , hq),

for every

R3(u,j,lj,hj) and R3(l,h) .-

nY=1 R3(j, lj, hj) . (R4) For this vector l = (h,.. ., lq) , there exists a coloring 'l/J E '11(l) such that the event R4(¢>, l) := { ¢>(e) = 'l/J} occurs. (R5) For these l, h, and 'l/J, if for j E [q], we define Aj = Aj('l/J) := { v E e : 'l/J(v) = 'l/J(Vj)} , then for every j E [q] and u E Aj, the event

R5('l/J,u,j,lj,h j ) := {(lj -l)s+ 1 ~ I(u) ~ (lj -l)s+h j} --

occurs. Let R5 ('l/J , f, h) := We conclude that

nj=l q

nUEAj-vj

F2(e,a,T,l)~ _U RI(l)n [_U £E82

n

U ?/JEiJ! (l)

.

R5('l/J, U,), lj, hj) .

(R2(l,h)nR3(l,h)

hE[s]q

(R4('l/J,l) n R5('l/J,l,h)))].

230

A . Kostochka, M. Kumbhat and V. Rod]

So, we estimate (19)

Pr(F2(e,a,T,l))

::; L

Pr(R1(i))

lEs2 x

L

Pr({R2(£:h) n R3(£:h) I Rl(i)})

hE[sjq

L

Pr({R4(1/J,f) I Rl(i) n R2(£:h) n R3(£:h)})

t/JE'IJ(l)

We now will gradually evaluate and simplify the expression in (19). We start from Rl(i) . Since the vertex sets ofT(ej) for distinct j are disjoint and by Lemma 10, for every j the event R1(j, lj) = W( ej, a-2Ij- 1, T(ej), lj -1) depends only on the values of I (v) and ¢( v) for v E V (T( ej)) , q

Pr(R1(i))

= rrPr(R1(j,lj)). j=l

If T( ej) has Yj edges, then by the induction assumption,

Thus,

( . . ) rr q

(20)

Pr Rl(£)::;

j=l

ED-Yj

(

-

r ) (n-Ij+1) /(n+1)

lnr

.

Now, consider the events R3(u ,j, lj, hj). Since all I(v) are independent and all trees Tj are vertex-disjoint, R3 (u, i.lj , hj ) does not depend on R1(j' ,lj') for all j' =J j. Furthermore, if I(u) > s(lj - 1), then W(ej ,a - 2Ir 1,T(ej),lj -1) does not depend on the particular value of I (u) in this range. So,

231

Coloring Uniform Hypergraphs with Small Edge Degrees

By the independence of I (u) for distinct u, similarly to (17) and the argument following (17), we have Pr ({ R3(j , lj, hj) I Rl(l)} )

~

I: Pr {M = M(ej,lj -I)} (1- p(hj -1))

r-IMI-l

~ cO.5r

MCe - J'

+

I:

Pr {M = M(hj -I)} (1- p(h j

-1)r-1M1-

1

Mt;e : IMI
Since cO.5r ~ 0.1(1 - psf ~ 0.1( 1-p(hj-1)) r- z , again by the independence of I(u) for distinct u, we conclude that q

(21)

Pr ({ R3(~ h) I Rl(l)} ) ~

II 1.1(1- p(h j -1)) r-z . j=l

Now, we consider events R2(j , lj , hj) = {I(vj) = s(lj - 1) + hj}. Let j E [q]. Similarly to R3(u,j, lj, hj), R2(j, lj , hj) does not depend on u,(j', lj') for all j' =1= i . and •

--+

Pr ( {R2(J ,lj,hj) I R1(f)}

)

~1

p lnr (l )~p(l+-). - ps j -1 r

Moreover, again by the independence of I (u) for distinct u, R2 (j, 1j , hj ) is independent of all other R2(j', lj', hj,) and of R3(~ h). Thus , since q ~ k « lnr,

and together with (21) , we obtain q

(23) Pr({R2(~h)nR3(~h)

I Rl(f)}) ~ 2pqII1.1(1-p(hj -l)r- z . j=l

Now for given ~ h, and 'l/J E w(l), we evaluate

232

A . Kostochka, M. Kumbhat and V. Rodl

Observe that each event RI(t) n R2(~ h) n R3(~ h) already fixes the colors of VI , . . . , "« in ¢, but all other vertices of e are "free". Since for each vEe, the value of ¢(v) is chosen independently of all other vertices, for each ~ h, and 7jJ E w(t), we have

(24) Finally, for given ~ h, and 7jJ E

w(f), consider the event R5(7jJ,~h):= {R5(7jJ,~h) I RI(t)nR2(~h)nR3(~h)nR4(7jJ,t)} .

For j E [q], let Aj = Aj(7jJ) := { vEe: 7jJ(v) = 7jJ(Vj)} and aj = IAjl- 1. By definition, for every j E [q] and every U E Aj - Vj, the probability of the event

R5(7jJ, u,j, lj, hj) := { (lj - 1)8 + 1 ~ I(u) ~ (lj - 1)8 + hj} is at most phj. Since the values of I (v) for vEe - {VI, .. . ,vq} do not depend on RI (t) n R2(~ h) n R3(~ h) n R4(7jJ, t), and are independent of each other for distinct v, q

Pr(R5(7jJ,~h)) ~

II (phj)a

j

•

j=I

Since each 7jJ E w(h, ... , lq) is completely defined when we choose disjoint sets Al - VI,... , A q - "« in e - {VI, . . . ,Vq},

L

(25)

P4(1j;,l,h)Pr(R5(7jJ,l,h))

'l/JE'l!(l)

Thus plugging (20), (23), and (25) into (19), we have

(26)

Pr(F2(e,a,T,l)):::;.?= £E8 2

(27)

x

q

(

)=1

_L 2pq (IT 1.1(1- p(hj -l)rhE[s]q

)=1

r ) (n-lj+1) /(n+1) )

(

pcD-Yj lnr

Z )

233

Coloring Uniform Hypergraphs with Small Edge Degrees

We now will simplify and estimate the expression in (27). First observe that Thus since 0

< p < 2- kTr and hj

::; S ::;

p(~'¥I)r' we have

1.1(1 - p(hj - 1)) r-z ::; 1.1(1 _ p)(r-z)(h j-l) ::; 1.2(1 _ p)(r-z)h j. So, the expression in (27) is at most 2

_:2: pq( IT 1.2(1- p)(r-Z)h j) hE[s]q

(28)

= 2k- r

)=1

tt

1.2p(1 - p)(r-z)h 1

(:J

k(p hl)a 1

hl=1 al=O

X

(29)

[h~l a~o 1.2p(1- p)(Nlh, (;2)k(Ph 2)a,

x[fl to

1.2p(1- p)('-'lh,

(;}(Ph

q )"' ] ..

For j = q, q - 1, . . . ,1 (in this order), we can estimate

E, t/

::; 1.2pk

t

(;J k(ph; t (:.)

2P(1 - pJi'-'lh,

(1 - pr hr zs s

::; 1.2pk(1- p)-ZS

(phj)a

aj=O

hj=1

)a, j

)

:2: (1- prhj (1 + phjr hj=1

J

234

A. Kostochka, M. Kumbhat and V. Rod] s ~

1.2pke(1-p)(n+l)r LJ (1- prhj (1 + prh j zlnr

"""

hj=1 z~r

< 1.2(ps)ke (l-p)(n+l)r

~

4~

1.2(ps)ke (l-p)(n+l) .

Since ps ~ r(~~I)' n + 1 ~ 2, and 4k 2 = -Inc:, we have

1.2(ps)ke(1

2

4k p)(n+l)

< 1.2 -

In r

r(n + 1)

()( ) 2/ kc:-1/ I-p n+l < e- 3k 2

kIn r • 2r(n + 1)c:

Plugging these bounds into (28)-(29), we obtain that the expression in (27) does not exceed

2k-r

n

(e-3k2/2

.

J=1

rr

kIn r ) < k- r 2r(n + 1)c: .

(e-3k2/2

J=1

kIn r ) . r(n + 1)c:

Thus, by (26)-(27)

(30)

Pr(F2 (e, a ,T , l))

rr

(..!-) n~~:l)

q

S """ k- r (C:D-Yj ~ . lnr £Ee 2 J=1

(e- 3k2/2 kIn r ). r(n+1)c:

Note that

k- r D-Yl-"'-Yq = k- r D-y+l S D-Yc:

(31)

C~ r) n~l

•

Hence (recalling that D = c:kr(l~r)n~l), we may rewrite (30) as

(32)

Pr(F2 (e, a ,T ,l))

~ D-Yc: (..!-) n~l ~L In r

Ii .

£E82 J=1

(c:

(..!-)~) In r

By the definition of 82, for every For every j =1= i' , we estimate

n-l+l

(33)

c: ( -

r )~

lnr

e-3k2/2

l

(e- 3k2/2 klnr ) . r(n + 1)c:

E 82 there is j* such that lj* = l.

klnr <--e k -3k2/2 <--. 1 r(n+1)c:-n+1 n+1

235

Coloring Uniform Hypergraphs with Small Edge Degrees

For j

(34)

= j*

we will gain more, since lj*

r ) n;;-~tl e-3k2/2 C (lnr

= l.

Namely,

kIn r <e -3k2/2 -k- (In-r) r(n+l)cn+l r

I

n+l

Plugging (33) and (34) into (32), we have

Pr (F2(e,a,T,l))

Since the summands in the last expression do not depend on the choice of l E 82 and 1821 ~ (1 + l)q ~ (n + l)q, we have

Pr (F2(e,a,T,1))

< D-Yc (~) n~l -

lnr

182

1

(n+l)q-l

(e-3k2/2_k_ (lnr) n~l) < n+l r -

This proves (13) for i = 2 and thus the lemma . _ Applying Lemma 11 for 1 = n , we get the following immediate consequence.

Corollary 1. Let e E E(G) and a E [k]. Let D := Ckr(l~r) n~l . Let T be a rooted sub-r-tree of G with root e. If T has y edges, then

236

6.

A. Kostochka, M. Kumbhat and V. Rodl

PROOF OF THEOREM

2

Recall the following version of the Local Lemma. Theorem 4 [1]. Let AI, A2, . .. , AN be any events. Let 8 1 ,82 , . .. , 8N be subsets of [n] such that for each i, Ai is independent of the events {A j : j E ([N] - 8i) } . If there exist numbers Xl, X2, . .. , XN E all i E [N], Pr (A) :::; Xi TI (1 - Xj), then,

[a, 1) such that for

ies,

Pr (

1\ iE[N]

II (1- Xi) > a.

Ai) 2:

iE[N]

Radhakrishnan and Srinivasan used it in the following form. Lemma 12 [6]. Let AI, A 2, . . . , AN be any events . Let 8 1,82, .. . , 8N be subsets of [N] such that for each i, Ai is independent of the events {A j : If for all i E [N], Pr (Ai) < ~ and

j E ([N] - 8 i )

}.

1\

> a.

Pr (

Ai)

l:

ses,

Pr (A j) :::;

i, then

iE[N]

Proof. We show that if the conditions of this lemma hold, then the conditions of Theorem 4 hold for Xi = 2Pr(A i), i E [N]. Indeed, with so defined Xi, inequality Pr (Ai) :::; Xi

II (1- Xj)

jESi

follows if

TI

(1 - Xj) 2: ~ holds. Furthermore,

j ESi

Hence by Theorem 4, we have the result.

•

Lemma 13. Let a < c:::; 4- kk-4 . If Pr (W(e ,a,T,n)) :::; cD-Y for every a E [k], every sub-r-tree T of G with y :::; k edges and for every e E E(T), then with positive probability, none of these events occurs.

237

Coloring Uniform Hypergraphs with Small Edge Degrees

Proof. Consider the probability space of the outcomes of Evolution. Let the events AI, .. . , AN be the events W(e , a, T, n) for all e E E(G), all a E [k] and all sub-r-trees T of G containing e with at most k edges. It is enough to verify that the conditions of Lemma 12 hold for our events AI, . . . , AN. Each of the conditions Pr (W(e, a, T, n)) < 1/2 immediately follows from Corollary 1. By Lemma 10, for the event Ai = W(e , a, T, n), we can take Si equal to the set of all events W(e/, 0', T', n) such that V(T') n V(T) #- 0. Now, fix an event Ai = W(e, a, T , n), where T has y edges, and estimate LjES; Pr(A j ) . Let W(e',a',T',n) E and suppose that the size of T' is y/. Then some edge e" of T' intersects V(T) (in particular, e" can be an edge of T , too). The number of ways to choose an edge that intersects V(T) is at most D + 1 if y = 1, and is at most yD, if y > 1. In any case, this number is not greater than kD. By Lemma 7, G contains at most (4D)Y'-1 r-trees of size y/ containing edge e". In each of such trees, there are y/ ways to choose a root, e', and k ways to choose the color o' , Since Pr (W(e' ,a',T',n)) :::; ED-Y', it follows that

s.

(35)

L Pr(A jES;

k

j ) :::;

k

L (kD) (4D)Y'-ly/k(ED-Y') = L k y'=l

2y/4 y'-lE

:::; k 44k - 1E.

y'=l

Since 0 < E :::; 4- kk-4 , the last expression in (35) is at most 1/4. Thus we are done by Lemma 12. • Now we are ready to complete the proof of the main theorem. Indeed, let G be a hypergraph satisfying the conditions of the theorem . Consider procedure Evolution. By Corollary 1, for each y-edge r-tree T, each edge e E E(T) and each a E [k], Pr (W(e, a, T, n)) :::; ED-Y. For k ~ 2, we have E = exp {-4k 2 } < 4- kk- 4 • So, by Lemma 13, with positive probability none of the events W (e, a , T ,n) occurs. It follows that in some outcome of Evolution none of the events W (e, a , T, n) occurs. By Lemma 9, in this outcome none of the events W(e, a, T, n) occurs. But then the resulting k-coloring will be proper.

238

A. Kostochka, M. Kumbhat and V. Rodl

REFERENCES

[1]

N. Alon and J. Spencer, The Probabilistic Method, Second Edition, Wiley (2000).

[2] P. Erdos and L. Lovasz, Problems and results on 3-chromatic hypergraphs and some related questions , in: Infinite and Finite Sets, A. Hajnal et al., editors, Colloq. Math. Soc. J. Bolyai 11 , North Holland (Amst erdam, 1975), pp. 609-627. [3]

A. V. Kostochka, Coloring uniform hypergraphs with few colors, Random Structures and Algorithms, 24 (2004), 1-10.

[4] A. Kostochka and M. Kumbhat, Coloring simple uniform hypergraphs with few edges, Random Structures and Algorithms, 35 (2009), 348-368. [5] A. Kostochka and V. Rodl , Constructions of sparse uniform hypergraphs with high chromatic number, Random Structures and Algorithms, 36 (2010),46-56. [6] J. Radhakrishnan and A. Srinivasan, Improved bounds and algorithms for hypergraph two-coloring, Random Stru ctures and Algorithms, 16 (2000), 4-32 .

Alexandr V. Kostochka

Mohit Kumbhat

University of Illinois, Urbana, IL , 61801 and Institute of Mathematics, Novosibirsk, 630090, Russia

Department of Mathematics, University of Illinois, Urbana, IL , 61801, U.S.A .

e-mail :

kostochk~ath.uiuc.edu

Vojtech Rodl Department of Math ematics and Computer Science, Emory University, Atlanta, GA 30322 , U.S.A. e-mail :

rodl~athcs.emory.edu

e-mail :

kumbhat2~uiuc.edu

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 239-286.

EXTREMAL GRAPHS AND MULTIGRAPHS WITH

Two

WEIGHTED COLOURS

EDWARD MARCHANT* and ANDREW THOMASON To Laszl6 Loiuisz on his sixtieth birthday

We study the extremal properties of coloured multigraphs H, whose edge set is the union of two simple graphs H; and Hi, (thought of as red and blue edges) on the same vertex set . Let 0 ~ P ~ 1 and let q = 1 - p. The extremal problem considered here, for a given fixed H , is to find the maximum weight pIE(Gr)1 +qIE(Gb)1 oflarge coloured multigraphs G that do not contain H as a subgraph. In fact , motivated by applications (typically to the study of hereditary properties by means of Szemeredi's Lemma), we consider the maximum restricted to those G whose underlying graph is complete - that is, every pair of vertices is joined by at least one edge. We describe some basic features of the extremal function in general; in particular it is shown that, for any class of forbidden graphs, the extremal function always has a finite description. We then look at some examples, including a detailed study of the case H; = K 3,3 and Hi; = H r = 2K3 . Our approach is that of classical extremal graph theory (and, as far as general properties are concerned, mirrors the work of Brown, Erdos and Simonovits on directed graphs). The recent work of Lovasz and his co-authors on graph sequences might offer an alternative approach.

1.

INTRODUCTION

A 2-coloured multigraph H is a multigraph whose edge set is the union of two simple graphs H; and Hi, on the same vertex set V(H) . Let Hu be the underlying graph of H; that is, Hu is a simple graph with vertex set V(H), "Research funded by Trinity College, University of Cambridge.

240

E. Marchant and A . Thomason

two vertices being adjacent in Hi; if they are adjacent in at least one of H; and Hi; We can think of H as being a colouring of the edges of the simple graph H u with three colours red , blue and green , an edge of H u having one of these colours according as it is in H; only, Hi, only or in both H; and Hi; The order of H, denoted by IHI, is defined as usual to be the number of vertices of H, namely IV (H) I. We say that a (large) 2-coloured multigraph G contains H , written He G, if there is an injection f : V(H) -+ V(G) for which f(x)f(y) E G; whenever xy E H; and f(x)f(y) E Gb whenever xy E Hi; In terms of simple graphs, this can be thought of as H u being a subgraph of Gu in such a way that the colours agree, except that the colour green is a "wildcard", meaning that a green edge of Gu can represent an edge of H u of any colour - red, blue or green . In particular, a complete graph of order n whose every edge is green contains every H of order at most n . (Notice here that we used the abbreviation "graph" for "2-coloured multigraph", as we shall do often.) We shall study the extremal properties of such 2-coloured multigraphs H . The appropriate measure, for our purposes, of the size of a 2-coloured multigraph G is obtained by giving each edge of G; the same weight p and each edge of Gb the weight q; the size of G is then the total weight. We normalize by assuming throughout that 0 :::; p :::; 1 and p + q = 1. Thus the size of a 2-coloured multigraph G is taken to be its p-weight wp(G), defined by

wp(G) = pe(Gr )

+ (1- p)e(Gb) =

pe(Gr )

+ qe(Gb)

where e(Gr ) is the number of edges in G; and e(Gb) is the number of edges in Gb. Equivalently, and this is how we shall usually think of it, wp(G) is the sum of the weights of edges in Gu , where red edges have weight p, blue edges have weight q and green edges have weight 1. It might seem that the natural extremal problem for a fixed 2-coloured multigraph H is to find, for each p and for each n, the maximum p-weight of a large 2-coloured multigraph G on n vertices that does not contain H . More generally, given a set 1i of 2-coloured multigraphs, one would find the largest weight of a G which contains no member of 1i as a subgraph. We shall indeed look at this general question in §3.6 and in §4. However, it it turns out to be more important for our main applications to restrict attention to those large 2-coloured multigraphs G which are complete, meaning that every pair of vertices in G is joined by an edge of some colour - in other words, Gu is complete. Therefore we define

kexp(1i,n) = max { wp(G) :

IGI = n,

G complete, H

ct G for all H E 1i} .

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241

The letter 'k ' in 'kex' here is to emphasize the completeness of the multigraphs G over which we are taking the maximum . Clearly, the unrestricted maximum, taken over all (not necessarily complete) graphs G, is at least as large as kexp (1l, n) and , as would be expected, it can sometimes be greater. Example 5.10 provides an instance of an 1l where the two maxima differ. All the same, in those cases when we are able to evaluate both maxima, they often agree. Our interest in kexp(1l, n) rather than the unrestricted extremal function is motivated by applications. However it is a fortuitous choice, because whilst most of the theory applies to both the restricted and the unrestricted functions, the strongest general results that we have, namely Theorems 3.23 and 3.25, are known only for kexp (1l, n). Section §3.6 discusses why the unrestricted problem is harder. We shall be interested mostly in single forbidden graphs H , and in this case we usually write kexp(H, n) instead of kex, ({H}, n). The general theory developed in §3 applies just as well to any class 1l (apart from the degenerate classes noted in §3.1). It is readily verified that kexp (1l,n)/ (~) decreases with n (c.f. Katona, Nemetz and Simonovits [27]) and therefore the limit

/'i,p(1l) = lim kexp(1l, n-+oo

n)(n)-l 2

exists. Since w p(G) :::; G) if IGI = n, we have 0 :::; /'i,p(1l) :::; 1. We shall study /'i,p(1l) rather than kexp(1l, n) itself; thus some of the fine detail of the extremal function is lost. Something can be said about this detail in some cases (see [31]) but in general the extremal situation can be quite complicated since, in particular, it contains the classical theory of ordinary undirected graphs (q.v. Bollobas [10] or Simonovits [38]) , which arises when H is blue-free and p = 1. By contrast, the extremal density /'i,p(H) is quite well-behaved. Complete multi-partite graphs are central to the study of extremal graph theory. The corresponding objects here are large complete 2-coloured graphs G whose vertices are partitioned into classes, such that the colour of an edge depends only on which classes its endvertices lie in. The description of which colours are used within and between the classes of G is given succinctly via the notion of a type, introduced in [12].

Definition 1.1. A type is a labelled 2-coloured complete multigraph, each of whose vertices is coloured red or blue.

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Hopefully it is clear that, by saying a type r describes a graph G, we mean that the classes of G correspond to vertices of r ; edges of G within a class have the colour of the vertex of r , and edges between classes have the colour of the corresponding edge of r. Associated with the p-weight wp(G) of G is a natural quantity called the p-value Ap ( r) of r (the definition is not needed yet but can be found in Definition 3.5). The significance of types, and the general way in which they describe extremal functions for 2-coloured multigraphs, is laid out in §3. The principal positive results of that section are Theorem 3.23 and Theorem 3.25. The second of these theorems states that, for any class 1l, there is some finite type r, such that no member of 1l is contained in a multi-partite graph that is described by r, and which satisfies /'l,p(1l) = Ap(r). Thus the determination of /'l,p(1l) comes down to finding a suitable type r. Theorem 3.23 is helpful here because it states that the search can be restricted to types of a quite specific form. The situation is not quite so straightforward as for classical undirected graphs, though; in §3.5 it is shown that there might be infinitely many types that satisfy Theorem 3.25, and that there need not be a finite subclass 1lo C H with /'l,p(1lo) = /'l,p(1l). A relationship between Ramsey games and 2-coloured multigraphs is described in §4, together with a conjecture akin to the Erdos-Stone theorem. The results of §3 are qualitative, and so the purpose of §5 is to make a quantitative study of /'l,p(H) for a few specific H's. Some simple examples have very straightforward associated types (this is always so when p = 1/2) but others are less straightforward. The case when H; = K3,3 and Hi, = H r = 2K3 is studied in some detail; this is one of the examples that motivated our work originally, and it exhibits some interesting and unexpected (to us) features. But it is high time that we offered some background to the whole matter, so we begin by describing the relationship of extremal 2-coloured multigraphs to Ramsey games, hereditary properties and edit distance. It is important also to discuss the work of Brown, Erdos and Simonovits, who made a study of the extremal theory of ordinary (uncoloured) multigraphs. Their work, and the additions provided by Sidorenko and by Rodl, underlies most of §3.

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2.

243

BACKGROUND

There are instances in the literature of studies lying near to the framework above . One of the best known works on the extremal theory of weighted graphs is that of Fiiredi and Kundgen [25] who, extending an initial investigation by Bondy and Tuza [13], determined the maximum total weight that a graph G of order n can have if the edge weights are integral and the total weight of the edges within any set of k vertices is bounded. Their results were applied by De Caen and Fiiredi [19] and by Keevash and Sudakov [28] in their work on the extremal hypergraph problem of the Fano plane. However these studies do not involve different types of edges distinguished by colours. One or two studies come closer to the present framework. Fiiredi and Simonovits [26], in their own solution of the extremal Fano plane problem, discussed the maximum weight of a 4-coloured multigraph containing no properly 3-coloured K 4 , when the colours all have the same weight. Still closer to home, Diwan and Mubayi [20] considered the largest size of a 2-coloured G where the forbidden 2-coloured subgraph H has no green edges . Their measure of the size of G is actually the minimum of er ( G) and eb( G), though during their investigation they do look at w p ( G) with p = 1 - 1/ (IHI- 1) . The extremal theory of 2-coloured multigraphs in precisely the form described above was (we believe) introduced by Richer [34], some of whose results are described later in §4. His motivation came from applications of Szerneredi 's Regularity Lemma in certain Ramsey-type games. We don't go into detail here, but the reader familiar with the Lemma will recognize the following situation (discussed in the survey of Komlos and Simonovits [29]) : when we wish to find an induced subgraph isomorphic to a fixed graph F in some very large graph, we apply Szemeredi's Lemma and obtain a reduced graph R, and then construct a 2-coloured multigraph G with V(G) = V(R) , the red/blue/green edges of Gu corresponding to uniform pairs of density close to one/close to zero/not close to either. Then, to find an induced Fin the large graph, it is enough to find a 2-coloured subgraph H C G, where H; = F and Hi, = F, the complement of F . In Richer's applications the very large graph is derived from some fixed graph together with a random graph on the same vertex set ; it is the edge probability p of this random graph that leads to the weighting wp . We shall say more about these applications in §4.

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2.1. The probability of hereditary properties

A graph property P is a class of (ordinary, simple, uncoloured) graphs closed under isomorphism. It is hereditary if it is closed under the taking of induced subgraphs, or, equivalently, under the removal of vertices. The relationship between hereditary properties and 2-coloured multigraphs is straightforward. Every hereditary property P can be described by a class of graphs F , such that the graphs in P are those with no induced subgraph in F (just take F to be the graphs not in P). Associate with each graph F E F the 2-coloured multigraph H where V(H) = V(F) , H; = F and Hi, = F ; thus Hu is complete and H has no green edges. Let 1l be the class of all these graphs H . We can write P = Forb(1l), implying that P is the property of forbidding members of 1l as induced subgraphs, though more precisely it is the members of F that are forbidden. Promel and Steger [33] determined the number of graphs in Forb ( {H} ) for a single graph H . The same was done for general hereditary properties by Alekseev [1] and by Bollobas and the second author [11], who further studied the probability that a random graph has the property P [12]. The latter results are stated in terms of the parameter ep(P) = limn-too Cn,p(P), where TCnA~) is the probability that a random graph of order n with edge probability p has the property P. The notion of a type (Definition 1.1) was introduced in [12] in order to describe ep(P) (though in very slightly different language) . The property P(T) consists of those graphs G for which V(G) has a partition into classes Vu , u E V(T), so that G[Vu] is complete or empty according as u is red or blue, and the bipartite graph with vertex classes G[Vu] and G[Vv] is complete or empty according as uv is red or blue (if uv is green then there is no condition on the bipartite graph) . It is shown in [12] that, for every hereditary property P , and for any

e > 0, there is a type T such that P(T) C P and Cp(P(T)) ::; cp(P) +c. Moreover it is shown that ep(P(T)) = Hp(T) , where Hp is an easily computed function (the notation H p was used because of a relation to entropy and it is not connected to the use in this paper of, say, Hi, as a graph). In this paper we use the somewhat more natural function Ap(T) of T (called the p-value of T; see Definition 3.5), which is essentially the same as Hp after a change of variable; to be exact , Hp = -(logp + 10gq)(1- ApI) where pi = (log q)/ (log p + log q). It turns out that we can say more than was said in [12] . By Theorem 3.25 applied to 1l where P = Forb (1l),

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there is a type r with P(r) C P for which Ap(r) = K,p(ll) . This has the consequence that Cp(P(r)) = Cp(P) . (The counterexample to the existence of such a r, asserted in [12], is erroneous .) Moreover it is possible to give some information about the structure of r itself, via Theorem 3.23. The precise application of the work in this paper to the probability of hereditary properties requires a little care, and is described in a separate note [32]. 2.2. Edit distance The notion of edit distance of relevance to coloured multigraphs is that studied by Axenovich, Kezdy and Martin [7], prompted by applications in biology, and by Alon and Stav [4], motivated by applications in computer science, especially in relation to property testing. Given a graph property P , the edit distance from a graph G to P is Dist (G, P)

= min {I E(J)6E(G)1

: J E P , V(J)

= V(G)}

and the edit distance from the class of all n-vertex graphs to P is Dist (n, P) = max {Dist (G,P) :

IGI =

n} .

In practice, only hereditary properties are considered. The crucial discovery of Alon and Stav is that, for any hereditary property P, random graphs G(n,p) have more or less the largest distance from P. More precisely, by developing the ideas of Alon and Shapira [3] , they show the existence of some probability p* = p*(P) for which Dist (n,P) = Dist (G(n,p*), p) + o(n2 ) holds almost surely. But their method gives no clue as to the value of p*. Nevertheless, the limit

ep(P) = lim IEDist(G(n,p),P) (n)-1 2

n-+oo

can be shown to exist (see [4]), where IE denotes expectation. It follows that ed (P)

= lim Dist (n,P) (n)-1 n-+oo

2

exists and that ed (P) = ep• (P) . Since ep(P) ~ ed (P) by the definition of edit distance, it follows that ep• (P) = maxpep(P); that is, p*(P) is the value of p that maximizes ep(P ).

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The relationship between edit distance and 2-coloured multigraphs can now be seen. Let F and 1£ be as in §2.1, so P = Forb(1£). Now choose a value of p and a complete 2-coloured G of order n, with H G for all HE 1£, and with wp(G) = kexp(1£ ,n). Given a random graph G(n,p) on the same vertex set as G, form J by deleting those edges in G(n,p) which are blue (not green) in G, and by adding those edges not already in G(n,p) which are red (not green) in G. If J contains a member of F as an induced subgraph then G must contain a member of 1£, bearing in mind that green edges of G can stand for edges of H of either colour. (Notice that the completeness of G is crucial here.) But G contains no member of 1£ and so J E P. The expected number of edge changes needed to form J is p times the number of blue (not green) edges plus q times the number of red (not green) edges; that is, G) - w p ( G) = G) - kexp(1£, n). This gives an upper bound on the expected distance of G(n,p) from P; hence ep(P) ::; 1-Kp(1£). The following proposition states that this inequality is tight; the assertion is implicit in [4] and explicit in Balogh and Martin [8, Theorem 11] (though stated there in different terminology).

ct

Proposition. Let P = Forb(1£) be a hereditary property. Then, for all p, ep(P) = 1 - K p(1£ ) holds. In particular, ed (P) = 1 - min, K p(1£ ). Note by Theorem 3.1 that K p (1£ ) is continuous in p and so the minimum exists. We sketch the reason why the proposition follows from Szemeredi's Regularity Lemma. As shown in [4]' Dist (G(n,p), p) is concentrated near its mean, so with high probability there exists J E P with IE(J)L:.E( G(n,p)) = ep(P)(~) + 0(n2). Take a uniform partition of J into k (a large number of) parts. The density of G (n, p) between each pair of sets in this partition will be close to p. So if we form the 2-coloured multigraph G whose vertices are the k parts and whose edges are red/blue/green according as the pairs have density in J close to zero/close to one/neither of these, then the red and blue edges of G indicate places where G(n,p) is far from J . In particular, with high probability IE(J)L:.E(G(n,p)) 2: ((~) -wp(G))(I)2+ o(n2). But G cannot

I

I

contain a member of 1£ for then, assuming all pairs of the partition are uniform (see below), we could find a member of F induced in J. Thus w p ( G) ::; kexp(1£ , k) = K p (1£ ) (~) + 0(1) . Therefore

2) ep(P) (;) + o(n 2: (1- K p(1£ )) which completes the sketch.

(~) (~r + 0(n2)

Extremal Graphs and Multigraphs with Two Weighted Colours

247

The reader familiar with Szemeredi's Lemma will have spotted a couple of holes in the above sketch. Most obviously, we assumed all pairs of the partition were uniform, which is known to fail in general. But admitting non-uniform pairs is equivalent to admitting 2-coloured G that are not complete, in which case the inequality wp ( G) ::; kexp(H, k) might fail. More subtly, if H is infinite, it is not possible to bound in advance the number of vertices of the graph F, and its construction in J requires a level of uniformity that depends on IFI. Both of these difficulties can be resolved by using a development of the Regularity Lemma due to Alon, Fischer, Krivelevich and Szegedy [2], see [4, Lemma 2.7], asserting the existence of an induced subgraph J' c J meeting each part in en vertices and, restricted to which, every pair is uniform, and in which the degree of uniformity is bound to the number of parts. The argument can then be carried through with J' in place of J . The proof, involving a random graph superimposed on a Szemeredi partition, is very similar to the work of Richer (see §4) except in the important detail that Richer needs only the ordinary Regularity Lemma, meaning that his investigation of extremal 2-coloured multigraphs includes non-complete G, although often in practice this makes no difference. The edit distance of a considerable variety of properties P is investigated by Axenovich, Kezdy and Martin [7] and by Alon and Stav [5], often by studying the associated types. One of the outstanding properties left unresolved by [5] is that of containing no induced K3,3. The edit distance of this property was subsequently determined by Balogh and Martin [8]. The corresponding extremal 2-coloured graph problem is particularly interesting and is examined in more detail in §5.4. 2.3. Directed graphs and multigraphs

The extremal theory for directed graphs and multigraphs was developed systematically in a series of substantial papers by Brown, Erdos and Simonovits [15, 16, 17]. An important further contribution was made by Sidorenko [36], whose theorems and counterexamples answered many unresolved questions, and another significant counterexample was supplied by Rodl and Sidorenko [35] . Brown and Simonovits have written a survey of the subject [18]. The object of all of these studies is as follows. In the multigraph situation, given a natural number m, some class of multigraphs .c is forbidden, and it is required to determine the maximum size of multi-

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graphs which contain no member of I: and in which the edge multiplicities are bounded by m. The directed graph situation is similar; multiple edges are allowed but no more than m in each direction between any two vertices. What is made clear in these papers is that the extremal theories for directed graphs with multiplicity at most m and for multigraphs with multiplicity at most 2m are very close in spirit, to the extent that results proved for one theory can be asserted for the other without real need for a second proof to be given. Nevertheless it is not possible to deduce any formal equivalence between the theories (there are of course occasional one-way implications that can be observed) . It is apparent from the methods used in these studies that they give a lot of information about the extremal properties of 2-coloured multigraphs, as we shall develop in §3. The nearest equivalent might seem to be the theory for multigraphs (and we shall not say anything more about directed graphs). Indeed, given the continuity of Kp(H) (Theorem 3.1) it would be enough to evaluate this function for rational values of p, in which case (by dropping the requirement that p + q = 1) we could assume the edge weights p and q were integers. It might then be hoped that the theory of 2-coloured multigraphs could be deduced from the ordinary theory of multigraphs with, say, m = p + q. However this is not the case because, crucially, in a 2-coloured multigraph we can distinguish the colour of the edges. Particularly important in the classical undirected multigraph theory is the case m ~ 2, which exhibits special features which either are unknown for the cases m ~ 3 or which do not in fact hold. It turns out that the theory for 2-coloured multigraphs exhibits a somewhat remarkable character, in that some, but not all, of the special features found in the classical case when m ~ 2 hold also for 2-coloured multigraphs. In that sense the coloured theory might be thought of as "mimicking the case m = 2!" of the classical multigraph theory. Let us be more specific. Turan's theorem, its extension by Erdos and Stone [24] and the developments by Erdos and Simonovits [21, 22, 23, 37] show that complete multipartite graphs are asymptotically extremal for any forbidden class I: of simple undirected graphs, and indeed that all extremal graphs have approximately this structure (the latter phenomenon is called stability). For 2-coloured multigraphs, the objects that play the role of multipartite graphs are the classes Q(T) described by types T (the 2-coloured version ofthe graph properties P( T): see Definition 3.2). An alternative way to specify the structure of the graphs in Q(T) is by means of a (symmetric)

Extremal Graphs and Multigraphs with Two Weighted Colours

249

matrix, whose rows and columns are indexed by the vertices of r and whose entries are p, q or 1 according to the weights of the appropriate edge colours. It can be shown without too much difficulty, perhaps either by Szemeredi's Lemma or, preferably, by a simple direct argument as in §3, that for any class 1-l, the extremal function /'i,p(1-l) can be approximated arbitrarily closely by the values /'i,p( Q(r)) where no H E 1-l is in Q(r) . The corresponding structures for multigraphs, in which the number of edges between groups of vertices is determined by a finite matrix, were analysed in depth by Brown, Erdos and Simonovits. They discovered several important properties of the case m :::; 2. In [15] they showed that there is a finite matrix that gives the exact limiting extremal density. Moreover in [17] they gave an algorithm to find all such optimal matrices (greatly simplified later by Sidorenko [36]). These optimal matrices must satisfy strong constraints [17] . Furthermore every class of forbidden multigraphs contains a finite sub-class whose extremal function is (asymptotically) equal to that of the whole class [17]. For the cases m 2: 3 the situation is quite different. In these cases it remains unknown whether there is an optimal matrix giving the exact extremal density. For m 2: 3 Sidorenko [36] has shown that there can be an infinite number of optimal matrices, killing any hope of an algorithm to find them or of proving a stability result for the extremal graphs. Finally, Rodl and Sidorenko [35] have shown that for m 2: 4 there need be no finite sub-class equivalent to th e whole class. Ideas used in both the cases m :::; 2 and m 2: 3 can be adapted for use here. So it is shown in Theorem 3.25 that for any class 1-l of 2-coloured multigraphs there is a type r that determines exactly the, alue of /'i,p(1-l). We also establish strong constraints on the nature of the .vpes involved (Theorem 3.23), which makes the evaluation of /'i,p(1-l) a realisic possibility in many instances. On the other hand, there might be inllnitely many optimal types in some cases. Likewise there need not be a fin.te sub-class H« C 1-l with /'i,p(1-lo) = /'i,p(1-l) . These are the reasons why we say that 2-coloured multigraphs lie somewhere between the undirected multigraph cases m :::; 2 and m 2: 3. An explanation of sorts for this phenomenon might be that in the case m :::; 2 the entries in the extremal matrices can have only two values, whereas more values are possible when m 2: 3. In the case of complete 2-coloured multigraphs, the matrix entries take three values p , q or 1, but the value 1 is not genuinely distinct, since it denotes green edges which are really the union of a red and a blue edge. Thus we acquire some of the properties of the case

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m ::; 2. On the other hand the above-mentioned counterexamples for the case m ~ 3 involve multigraphs of only three different multiplicities, and (for suitable p) their properties can be emulated here also.

2.4. Graph sequences

We remark finally that a new approach to classical extremal graph theory and related extremal problems has emerged recently via the study of graph sequences and their limits. Perhaps this approach could be fruitful in the investigation of 2-coloured multigraphs; in particular, LOV8,sZ and Szegedy [30] have re-proved the main results of [4] by this method. However, we have not pursued this idea ourselves. We move on now, at last , to our results.

3.

GENERAL RESULTS

Following some simple observations about K p in §3.1, we discuss the basic properties Q(T) in §3.2. As stated already, these were introduced in [12] and are analogous to the matrix properties of [15, 16, 17]. In particular, we shall describe extensions of properties and show how all properties can be approximated by basic properties. The remaining sub-sections contain results which are not so near the surface. In §3.3 it is shown that only certain special kinds of types need be considered in relation to extremal 2-coloured multigraphs (Theorem 3.23). The main result is Theorem 3.25 in §3.4, that for every class 1-£ there is a type T that determines exactly the value of K p (1-£ ). These results are modelled on those of Brown, Erdos and Simonovits [15, 16, 17]. The next sub-section contains examples, based on those of Sidorenko [36] and of Rodl and Sidorenko [35], to show that extremal sequences need not exhibit stability, nor need there be a finite subclass 1-£0 C 1-£ such that K,p(H o) = K,p(H) . Finally, in §3.6, we look at to what extent the results of this section carry over to the more general extremal situation for 2-coloured multigraphs, in which the underlying extremal graphs need not be complete .

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Extremal Graphs and Multigraphs with Two Weighted Colours

3.1. Continuity and convexity In the introduction we defined K,p(H) and stated that K,p(1l) was defined analogously. However we must exclude classes 1l which contain both a 2coloured graph with no red edges and another graph with no blue edges. In this case, Ramsey's theorem means that any sufficiently large 2-coloured G will contain a member of 1l. We call such a class trivial and , from now on, unless stated otherwise, we assume that all classes 1l under consideration are non-trivial; that is, if 1l contains any monochromatic graphs then these are all of the same colour. It is immediate from the definition that 0 :::; K,p(1l) :::; 1 for all P and all1l . Moreover, if no graph in 1l is monochromatic then K,o(1l) = K,1(1l) = 1, as demonstrated by monochromatic extremal graphs. On the other hand , if 1l contains a subclass 1lm of monochromatic graphs, say, red ones, then K,o(1l) = 1 and K,1(1l) :::; limn -+ooex (n, 1lm)/G) where ex(n, 1lm ) is the ordinary uncoloured extremal function for 1lm .

Let 0:::; PI < P < P2 :::; 1, so that P = XPI +(1-x)P2 for some 0 < x < 1. As mentioned in the introduction, kexp(1l , n)1 (~) decreases with n, so for each n we can pick G" containing no member of 1l such that IG" I = n and wp(Cn) ~ K,p(1l)(~). Now wp(cn) = pe(G~) + qe(Cb) = xWp1(Gn) + (1 - x )wp2(C n ). So XK,Pl (1l) + (1 - X)K,P2 (1l) ~ limn -+ oo ((xw p1(Gn) +

(1- x)w P2(cn)))/G) ~ K,p(1l). So K,p(1l) is convex, and thus continuous for 0 < P < 1. In fact it is continuous at the endpoints too ; for clearly kexpl ( l l , n ) ~ kexp(ll,n) -Ip - p/l(~), and so lK,p(ll) - K,pl ( l l ) Ip - p/l for all P, p' . Hence we have the following result (noted also in [4] and [8]) .

I : :;

Theorem 3.1. The function K,p(1l) is convex and continuous as a function of p.

One useful consequence of convexity is that a local minimum is a global minimum (as needed for edit distance §2.2).

3.2. Basic facts Recall the notion of a type that was given in Definition 1.1 in §1. These are the same as the coloured regularity graphs of [4] and [8]. Here is a further definition from [12].

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Definition 3.2. Let T be a type. The graph m x T is the complete 2-coloured graph G of order mlTI, whose vertex set has a partition {Vv : v E V(T)} with IVv I = m, such that edges within Vv are the same colour as v and the edges between Vu and Vv are the same colour as uv, for all u ,VET. The basic property Q(T) consists of all complete 2-coloured graphs G that are subgraphs of m x T for some m . Thus graphs in Q(T) are those having a vertex partition { Vv : v E V (T) } into classes of possibly different sizes, whose edge colours are the same as those in the definition of m x T except in the case of a green edge uv, in which case the subgraph between Vu and Vv must be complete but can be coloured in any way (and need not be monochromatic). This definition can be seen to be the 2-coloured equivalent of the property P(T) given in [12] and discussed in §2.1.

Definition 3.3. The 2-coloured graph H is said to be T-colourable if it is a subgraph of some member of Q(T) . The only difference between members of Q(T) and r-colourable graphs in general is that the latter need not be complete . The largest order of a type T for which H is not T-colourable is called the colouring number of H . This is equivalent to the parameter defined in regard to induced subgraphs by Promel and Steger [33] ; its relationship to edit distance was studied by Axenovich, Kezdy and Martin [7] . For the purposes of calculations about a type it is helpful to associate a matrix with it .

Definition 3.4. The weight wp(uv) of the edge uv of a 2-coloured graph is p, q or 1 according as uv is red, blue or green. The weight wp(u) of the vertex u of a type T is p or q according as u is red or blue. The p-matrix Wp(T) of the type T is the symmetric matrix whose rows and columns are indexed by V(T) ; the u, u entry is wp(u) and the u, v entry is wp(uv). Evidently T determines Wp(T), and vice-versa if p i= O'~' 1; we could, then, by and large, continue the discussion in terms of matrices rather than types, which is of course the approach of Brown, Erdos and Simonovits [15, 16,17] .

Definition 3.5. The p-value of the type T is Ap(T) = max z E.6. ZtWp(T)Z, where !:1 = !:1( ITI) is the simplex {z E [0,1]ITI : Zl + ... + ZITI = I} .

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Extremal Graphs and Multigraphs with Two Weighted Colours

Remark 3.6. This definition is prompted by the weights of members of Q(r) . For suppose 0 E Q(r) is such that 101 = nand {Vv : v E V(r)} is a partition of V (0) proving membership. The weight of 0 will be greatest if all edges between Vu and Vv are green whenever uv is green. Let IVvl = Yvn, so that Y = (Yl" ' " Y\T\) E ..6.. Then the p-weight of 0 is n) w p (0) -- p "(YV L.J 2 v is red

+p

n) (Yv2

+ q "L.J v is blue

L uVEE(Tr )

YuYv n 2 + q

L

YuYv n 2

uvEE(Tb)

Given any fixed z E ..6., we can choose 0 so that the entries of Y - z are O(l/n), so it follows that the maximum p-weight of graphs in Q(r) is Ap(r) G) + O(n) . The central point of the present paper is that we can determine /'i,p(1l) in terms of Ap(r) for those r such that the graphs in 1l are not .r-colourable. So it is worth stating the easy part of this relationship here , which follows from the fact that the graphs 0 E Q(r) just defined contain no member of 1l.

Observation 3.7. Let 1l be a class of 2-coloured multigraphs and let r be a type such that no graph in 1l is r-colourable. Then Ap(r) :S /'i,p(1l) . Our final definitions in this section are these.

Definition 3.8. An extension of the type r is a type a- wit h rea- and la-I = Irl + 1. Definition 3.9. A graph sequence is a sequence (on)~=l of 2-coloured complete graphs with IGnl ~ n. Let r be a type. We say that (on) contains r if for every m there is an n such that m x r is a subgraph of on. Notice that this definition is equivalent to stating that every r-colourable graph is a subgraph of infinitely many G",

Definition 3.10. The degree of the vertex v in the 2-coloured graph 0 is dp(v) = pl{u : vu E E(G r ) } +ql{u : vu E E(Ob)} The minimum degree is Op(O) = minvEV(G) dp(v).

I

I.

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The relevance of extensions to this work is provided by the next lemma. Lemma 3.11. Let (C n ) be a graph sequence and let

Let 7 be a type contained in (en). Let x = (Xl,' .. , Xlrl) E.6.. Then (C n) contains an extension CT of 7 for which

L

xuw(uv) ~

s,

uEV(r)

where {v} = V(CT) - V(7). Remark 3.12. The purpose of the vector X in this lemma is worth pointing out. It will be of use when we come to apply the lemma to specific examples in §5. Suppose we have a type 7 contained in (C n ) , and there is some subset of the vertices of 7 such that we would like an extension of 7 with the new vertex joined by green to this subset. Now the lemma provides an extension such that the weighted sum of all edge weights at v is large. To encourage the lemma to provide an extension of the kind we want, we can choose an x for which Xu is smaller for vertices u of less interest, so the contribution of the vertices of interest is amplified relative to the others. The definition of an extension is analogous to that of an augmentation which plays a fundamental role in the work of Brown, Erdos and Simonovits[15, 16, 17]. The idea is a very general one and applies equally to directed graphs, to multigraphs and to 2-coloured multigraphs (as remarked in [18], "augmentation does not feel the orientations") . Their definitions are in terms of the corresponding p-matrices Wp(7) and Wp(CT). The only essential difference between the two definitions is that in the definition of an augmentation the vector x is chosen to be one for which Ap (7) = Xt Wp(7)X . This difference is of no significance for the proofs of the general results but we retain it for its use in §5, as just described. Proof. If ~ = 0 it is enough to form CT by picking some vertex u E 7, adding a new vertex v of the same colour as u, joining it to u by an edge of that colour and joining v to every other vertex in the same way that u is joined. For then m x CT C 2m x 7 so CT is contained in (en) , and the final condition of the lemma is vacuous when ~ = O. So we may assume ~ > O. We shall show that, for each natural number m, there is some extension CT of 7 satisfying the conditions of the lemma and with m x CT c en for

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some n . This will be enough to complete the proof, because there are only finitely many extensions of 7, and so there is some extension (J for which the inclusion m x (J C G" occurs for infinitely many m, and so for all m (since m x (J C m' x (J if m < m'). So let m be fixed. Fix also 0 < e < 0/4. It now suffices to find an extension (J with m x (J C G" for some n, such that (J satisfies the weaker condition xuw(uv) 2:: 0 - 4e.

L

UEV(T)

This is because there are only finitely many extensions (J, and so some (J must satisfy the weaker condition for every e > 0, and hence must satisfy the condition of the lemma. Pick an integer M such that M > 6ml71c 1 and also lxuMJ - 3m > xuM/2 for every u E V(7) with Xu i= O. Now M x 7 C G" holds for infinitely many n ; pick such an n that satisfies n 2:: 2R(m)e-13IT1 (~) ITI , where R(m) is the two-colour Ramsey number for K m . By increasing n if necessary we may also assume op(C n ) 2:: (0 - e)ICnl. We shall show there is an appropriate extension (J of 7 with m x (J C G", Choose a copy of M x 7 C G" and denote its vertex classes by Vu , u E 7 . For each u, select a subset Xu C Vu with IX uI = lxuM Jand let X = Uu Xu. Thus M -171 ::; IXI ::; M < en/4. Given a vertex z E V(C n) and a set Z C V(C) let dz(z) = L,YEZ wp(zy) . Then L ZEV(G) dx( z) = LYEX dp(Y) 2:: dx(z) 2:: (0 - 2e)IXI} . Then

IXI(o -

e)ICnl::;

L

IXI(o - e)IGnl· Let

Y

= {z 1:-

X

dx(z)

z EV (G )

::; (IYI + IX!) IXI + (Icnl-IYI-IXI) (0 - 2e)IXI· Thus

IYI + IXI 2:: eICnl/(l - 0 + 2e) 2:: en and so IYI 2:: en/2.

Fix for the moment a vertex Y E Y. We claim that, for each u E 7, there is a subset Uu C Vu with IUul = m, so that the edges between Y and Uu are all the same colour, whose weight we denote by wu; moreover, if Xu i= 0, we claim Uu may be chosen so that W u 2:: dx,,(y)/IXul- e/X u I71. If Xu = 0 the claim is verified simply by choosing m vertices in Vu joined to y by edges of the same colour, and this is possible since lVul = M > 3m. If Xu i= 0,

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observe that if the claim fails then there are less than 3m edges between y and Xu whose weight satisfies the bound for wu; thus

dx u (y) ::; 3m + (IXul- 3m) {dx u (y)/IXul- c/xuITI)

< 3m + dx; (y) - (IXul- 3m) c/xulTI < 3m + dx..{Y) - x uMc/2xulTI < dx; (y)

by the choice of M

also by the choice of M.

This contradiction establishes the claim. By the definition of Y we have dx(Y) ~ (0 - 2c)IXI. Moreover IXI ~ M -ITI, and so dx(Y) ~ (8-2c)M -ITI ~ (8-3c)M. Since IXul ::; xuM , the claim now implies L:u XuW u ~ L:u (dx..{y)/M - c/ITI) = dx(y)/M - c ~ 8 - 4c. Let U(y) = Uu Uu. There are at most (~)ITI possible choices for this subset of Uu Vu ' There are at most 31T1possible ways that the edges between

y and U(y) can be coloured. Now WI ~ cn/2 ~ R(m)3ITI{~)ITI. So there is a subset Z C Y with IZI = R(m), such that the sets U(y) are identical for all y E Z and moreover every y E Z is joined in the same way to U(y). By the definition of the Ramsey number we can pick a subset of m vertices in Z that span a complete red or a complete blue K m (in this instance a green edge is thought of as containing both a red and a blue edge). Call this subset Uv. Then U; together with UUEV(T) Uu is the vertex set of m x (5 for some extension (5 of T, where {v} = V ((5) - V (T). Moreover, by construction, we have L:UEV(T) xuw(uv) = L:u XuWu ~ 8 - 4c. Lemma 3.13. Let T be a type and let x E .6. satisfy Ap(T) = XtWp(T)X. Let (5 be an extension oi t such that L:UEV(T)XUW(uv) ~ 0 > Ap(T), where {v} = V((5) - V(T) . Then Ap((5) ~ 82/{28 - Ap(T)) > Ap(T). Proof. Given 0::; ~::; 1, let y E [0,1]10- 1 satisfy Yu = ~xu ifu E V(T) and Yv = 1-~ where {v} = V((5)- V( T). Then y E .6. { 1(51) so Ap((5) ~ ytWp((5)Y . Thus Ap((5) ~ eAp(T) + 2~(1- ~)8. Taking ~ = 8/{28 - Ap(T)) gives the desired result . _ A standard lemma relates the minimum degree condition of Lemma 3.11 to the p-weight condition involved in the definition of /'i,(1l).

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Lemma 3.14. Let 1£ be a class of 2-coloured graphs. Then there is a graph sequence (Gn) with liminfn~oo (8p(Gn)/IGnl) = K,p(1£) , such that no G" contains a member of 1£. Proof. It suffices, for each n , to find a graph G" with at least n vertices , with minimum degree at least (K,p(1£) - 3/n) IGnl , and which contains no members of 1£. This sequence will satisfy liminfn~oo (8p(Gn)/ IGnl) 2': K,p(1£) , and it is impossible that the inequality is strict, for then we would have a sequence with liminfn~oowp(Gn)/(~) > K,p(1£) , contradicting the definition of K,p (1£) .

To find G" we note that, by the definition of K, = K,p(1£) , there is a some N > n2 and some graph GN of order N with wp(G N ) 2': (K, -1/n)(~) and containing no member of 1£. Construct a maximal sequence of graphs GN ~ GN-l ~ ... ~ GM such that IGjl = j and Gj - 1 is obtained from Gj by removing a vertex of degree at most (K, - 2/ n) (j - 1) in Gi- Then M > n, for otherwise Gn appears in the sequence, and

which is impossible. Since the sequence stops at G M it follows that Op(GM) 2': (K, - 2/n)(M - 1) 2': (K, - 3/n)M, so we may take G" = CM. • The following approximation theorem is the 2-coloured analogue of [16, Theorem 2J and , given the equivalence of K p (1£ ) to other parameters as discussed in §1, can be found in [5 , Lemma 3.4], [8, Theorem 11J and [12 , Theorem 1.1J. Theorem 3.15. Let 1£ be a class of 2-coloured graphs. Then K

p(1£ ) = sup { >'p( T) : no graph in 1£ is T-colourable} .

Remark 3.16. This theorem is made redundant by the deeper Theorem 3.25 appearing later, in which the supremum is replaced by a maximum. Proof. Let T = {T: no graph in 1£ is T-colourable}. Let s = sup { >'( T) : T E T} and K = K p (1£ ). Observation 3.7 shows that s ~ K; the point of the theorem is that equality holds.

Suppose on the contrary that s < K. By Lemma 3.14 there is a graph sequence (cn) with liminfn~oo (o(cn)/Icnl) = K such that no G" contains

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any member of 1-l. Ramsey's theorem implies that the sequence (Gn) contains a type Tl with a single vertex. By Lemmas 3.11 and 3.13 we can find a sequence of types Tl C T2 C .. . C Tt, all contained in (Gn) , such that Ap(Tj+l) 2:: K,2/ (2K, - Ap(Tj)) . From this it follows by induction that Ap(Tj) 2:: K,(1 - Ijj) . Let t = K,/(K, - s)l + 1; then t > K,/(K, - s) and so Ap(Tt) > S. By the definition of s, some graph HE 1-l is Tt-colourable, which is to say that Hem x Tt for some m. But (C n ) contains Tt and so there is some n for which m x Tt C G", Thus H C G", contradicting the definition of the sequence (C n ) and completing the proof. _

r

3.3. Core types

Clearly, if T is a type, then Ap(T') :s; Ap(T) for every sub-type T' of T (by extending vectors z' E i1( IT'I) by zeros to vectors z E i1( ITI)). It might be that there there is some T' with Ap(T') = Ap(T). In contexts such as that of Theorem 3.15, where it is only the p-value Ap(T) that matters, rather than T itself, it is evident that we need not consider T but only T' . This observation leads naturally to the next definition. Definition 3.17. Let 0 ::; p ::; 1 be given. A type Ap(T') < Ap(T) for every proper sub-type T' of T.

T

is called p-core if

Remark 3.18. Note that a typ e that is p-core for some p might be pcore for every p but need not be. For example, let T(O , k) be the type consisting of k blue vertices joined by green edges (for the more general T(a, b), see Definition 5.1). It is easily seen that T(O, k) is p-core for all non-zero p ; indeed the optimal vector assigns weight 1/k to each vertex and Ap ( T(O, k)) = 1 - p/k. On the other hand , the type of order two consisting of two red vertices joined by a blue edge, is p-core for p < 1/2 but not for p 2:: 1/2.

Second, note that the definition can be stated equivalently in terms of the p-matrix Wp(T) of T, which is the approach of [15,16,17] . Indeed, Tis p-core if and only if the matrix Wp(T) is den se in the sense of those papers. Lastly, note that this definition differs from that in [12] but only in a change of variable (as explained in §2.1 and in [32]) ; that is, for each p, there is a p', such that T is p'-core if and only if the corresponding type is p-core in the sense of [12].

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On contemplation of the definition, some useful properties of p-core types reveal themselves, and they have been discovered more than once. Consider an optimum vector x E ~ such that xtWx = Ap(T), where W = Wp(T). In a graph in Q(T) of maximum p-weight, the class sizes are proportional to the entries of x (q.v. Remark 3.6). But the degrees in a graph of maximum pweight are evidently more or less equal (if not, move a vertex from one class to another, as done in [12, Lemma 5.3], [15, §2]). This means Wx = -Ap(T)e, where e is the all one vector. We make note of this fact.

Fact 3.19. Let T be a p-cote type with p-weight A = Ap(T). Let x E ~ satisfy xtWx = A, where W = Wp(T). Then Wx = Xe. In other words, xuwp(u) + 2:uvEE(T) xvwp(uv) = A holds for all U E V(T) . The definition of p-core means that no co-ordinate of x can be zero. Thus by shifting weight from one vertex to another it can be seen that T has no twin vertices ([12, Lemma 5.2], [15, §2]). Indeed, if wp(uv) ~ wp(u) = wp(v), define the vector y by Yu = -xu, Yv = xu, Yi = 0 otherwise; then yte = 0 and Wy = O. Let z = x + y. Then z E ~ and ztWz = xtWx + 2ytWx + ytWy = ..\, giving an optimal vector z with a zero co-ordinate, a contradiction. We note this fact too .

Fact 3.20. Let T be a p-cote type and let u and v be two vertices with wp(u) = wp(v). Then wp(uv) > wp(u). The converse of Fact 3.19 clearly holds also, since if T is p-core and W x = .\e then x is optimal. So convex linear combinations of optimal vectors are themselves optimal, and hence there must be a unique optimum (else we could create one with a zero entry). Then W is invertible (else we could perturb the optimum optimally), so x = .\W-1e and .\ = l/e tW- le ([17, Lemma 2]' [8, Lemma 10]). However, we make no use of these properties of p-core types. Neither shall we use Sidorenko's elegant characterization of their matrices [36] , but it bears repeating all the same.

Proposition 3.21 (Sidorenko [36]). A type p-matrix W = Wp(T) satisfies

T

is p-cote if and only if its

• W is non-singular and all components of W-1e are positive, and • ytWy < 0 for every non-zero vector y with ety = O. The next lemma will be a very useful tool in handling p-core types, despite its rather clumsy statement.

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Lemma 3.22. Let r be a p-core type where p ::; 1/2 and let x E ~(Irl) satisfy xtWp(r)x = Ap(r). Let B be the set of blue vertices of r, and suppose that all the edges of r incident with B are green. Let 0' be an extension of r such that L:uEV(T)XUW(uv) > Ap(r), where {v} = V(O')-

V(r) . Suppose now that -r' is a p-core sub-type of 0' with Ap ( r') = Ap ( 0'). Then Be V(r'), and all the edges of r' incident with B are green. Proof. Let b E B . Fact 3.19 implies that xbwp(b) + L:buEE(T) xuw(bu) = Ap(r). But w(bu) = 1 for all bu E E(r). SO L:UEV(T) xuw(uv) > Ap(r) is possible only if w(bv) > w(b), that is, the weight of the edge bv is greater than that of the vertex b itself. But b is blue and q ~ 1/2, so bv must be green. Thus b is joined by green to every other vertex in 0' . Hence every vertex of 0' incident with B is green. It remains to show that B C V(r'). Now by Lemma 3.13, Ap(r') = Ap(O') > Ap(r). Since r is p-core this means r' cannot be contained in r and so v E V(r'). Suppose that some vertex b E B is not contained in r', Clearly the type r' - v + b, which is a sub-type of t , must have p-value at least that of r', because b is blue and is joined to every other vertex by green. But this would again contradict r being p-core: thus b E V (r'). • The next theorem, which is of practical significance, is the main result showing that the edge colours in a p-core type are very restricted; they must be green, unless they join two vertices of the same colour and this colour has the smaller weight. The proof is based on that of Theorem 5 in [17J (announced in [15, Theorem 3]), which is the analogous theorem for undirected multigraphs with edge multiplicity m ::; 2. A suitable analogue for higher multiplicites is unknown, so this is one of the properties of multiplicity m ::; 2 multigraphs that holds for 2-coloured multigraphs, as discussed in §2.3.

Theorem 3.23. Let r be a p-core type. Then all edges of r are green, apart from • if p or

< 1/2, when some edges joining two red vertices might

• if p

> 1/2, when some edges joining two blue vertices might be red.

be blue,

Remark 3.24. The fact that all edges must be green when p = 1/2 accounts for the simplicity of the study of random hereditary properties with edge probability 1/2 in [l1J as compared with the general case in [12J .

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261

Proof. By Fact 3.20 and by symmetry, it is enough to assume that p < 1/2 < q and to show that all edges meeting a blue vertex are green. Let B be the set of blue vertices of 7 . Certainly the edges within B are green, by Fact 3.20. Let Po be the sub-type of 7 spanned by B. Then Po = 7(0, IBI) (see Remark 3.18) so Po is p-core. We shall show that, given a p-core proper sub-type p C 7 which contains B and in which all edges incident with B are green, then there is another p-core sub-type p' C 7, also containing B and in which all edges meeting B are green, and such that Ap(p) < Ap(p'). Applying this process repeatedly, starting with Po , we generate a sequence of sub-types of strictly increasing pvalue, which must therefore reach 7 after a finite number of steps , completing the proof. Let p be a p-core sub-type as described and let x E ~(Ipl) be an optimum vector. Let (C n ) be a sequence of graphs in Q(7) of maximal weight; then liminf (bp(cn)/Icnl) = Ap(7) (see Remark 3.6). Since 7 is pcore, and no co-ordinate of an optimum vector is zero, the construction of G" shows that the sequence (C n ) contains 7 , and hence it certainly contains p. If p =1= 7 , then Lemma 3.11 applied to p shows that (C n ) contains an extension a- of p with L:uEV(P) xuw(uv) ~ Ap(7) > Ap(p), where v is the vertex added to p to make a, Let p' be a p-core sub-type of awith Ap(p') = Ap(a-) . Then Lemma 3.22, applied to p and p', shows that B C V(p') and that all edges of p' incident with B are green. All that remains is to show that p' is actually a sub-type of 7 . Now ais contained in (C n ) and so p' is also. In particular, (171 + 1) x p' C G" for some n . Denote the classes of (171 + 1) X p' by Uu , u E V(p'). By the construction of (C n ) we may pick, for each u, a vertex f (u) E V (7) such that two vertices of Uu lie in the same class Vf(u) of G" , But then u must have the same colour as f(u). If now f(u') = f(u) for some other vertex u' E V(p'), then the edge uu' is the same colour as both u and u' , contradicting Fact 3.20. So the map f : V(p') -+ V(7) must be injective, and this means p' C 7. • Theorem 3.23 comes in very handy when calculating K,p(H) for 2coloured multigraphs H, as we shall see when we look at examples in §5.

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3.4. Exactness

We are now in a position to prove the principal positive result of §3. This corresponds to the main theorem of Brown, Erdos and Simonovits [15] for multigraphs with multiplicty m ::; 2, on whose proof the proof here is based. Theorem 3.25. Let 1l be a class of 2-coloured multigraphs and let p ::; 1. Then there is a p-core type l' such that no H E 1l is l' -colourable, and That is,

°::; Ap(r) = /'i,p(1l).

/'i,p (1l) = max { Ap(1') : no graph in 1l is l' -colourable} . Remark 3.26. It is possible that infinitely many types satisfy the conditions of the theorem, as will be shown in §3.5. Proof. By symmetry, we may assume p ::; 1/2. Select a graph sequence (Cn) as given by Lemma 3.14, with liminfn -+ oo (b'(cn)/Icnl) = /'i,p(1l) , such that no G" contains a member of 1l. If (J is a p-core type contained in (Cn ) then, by Theorem 3.23, the blue vertices of (J are joined to each other by green edges. So if there are k of these vertices then they span the type r(O,k) (as mentioned in Remark 3.18). If HE 1l then H is 1'(0, IHI)colourable so k < IHI. Hence the number t = max {k : 1'(0, k) is contained in (C n ) } exists and is finite. The theorem will be proved if we show that there is some l' contained in (Cn ) with Ap(r) = /'i,p(1l). We shall assume, then, that no such l' exists , and derive that 1'(0, t + 1) is contained in (Cn ) , a contradiction that completes the proof. Let l' be a p-core type contained in (Cn ) such that 1'(0, t) Cr . Choose x E ~ so that xtWp(r)x = Ap(r), and apply Lemma 3.11 to obtain an extension (J of l' that is contained in (Cn ) . Let 1" be a p-core sub-type of (J with Ap(r') = Ap((J) . By Lemma 3.13, Ap(r') > Ap(r), and by Lemma 3.22, 1'(0, t) C 1" . Since we never come across a l' with Ap(r) = /'i,p(1l), this means we can construct an infinite sequence 1'(0, t) = 1'0 ,1'1 ,1'2 , . . . of p-core types , each contained in (Cn ) , each containing 1'(0, t) , and of strictly increasing p-value; in particular they are distinct and so their orders are unbounded.

Let m be a positive integer. Choose some Ti in the sequence with ~ t + m and let p be a sub-type of rj containing 1'(0, t) plus mother vertices. By Theorem 3.23 applied to rj, all the edges of p meeting 1'(0, t) are green, and the other edges are blue or green. Since p is contained in (C n ) , there is some n with m x p C G", Now the t classes of m x p corresponding

Irjl

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263

to the sub-type r(O, t) C P give a copy of m x r(O, t); by picking mother vertices, one in each of the m remaining classes of m x p, we obtain m further vertices joined to each other by blue or green edges and joined to m x r(O, t) by green edges. This construction shows that m x r(O, t + 1) em x p. and so m x r(O,t+ 1) C G", Therefore r(O, t + 1) is contained in (Gn), and we are done. _

3.5. Stability and Finiteness Unlike the analogous theorem for multigraphs with multiplicity m :s 2, there can be an infinite number of types that satisfy the conditions of Theorem 3.25. To show this, we adapt an example of Sidorenko [36], constructed to show the corresponding fact for multigraphs with m 2: 3.

Theorem 3.27. Let 1 2: 4 be an integer and let p = 1/1. Let H be the complete two-coloured multigraph of order 1+ 1 consisting of a red star KIl, with all other edges being blue. Then there are infinitely many p-cote types r such that H is not r-colourable and Ap(r) = Kp(H). Proof. It will be shown later in Example 5.5 that Kp(H) = 1 - 1/1. If r is any type whose vertices are red and in which th e green edges form an (1 - 2)-regular graph, the other edges being blue, then H is not rcolourable . Theorem 5.6 shows that any such r for which the green edges form a connected graph is p-core when p = 1/1, and Ap ( T) = 1-1/1 = Kp (H).

-

Thus there is no hope of proving a stability result along the lines that an extremal sequence for H must contain graphs whose structure is close to one of a finite number of possible structures. We mention, though, that Alon and Stav [6] have proved certain stability results related to KI /2(1-l.). Another finiteness property that holds for multigraphs with multiplicity m :s 2 is that every family contains a finite subfamily with the same extremal density [17, Theorem 3] . This too fails for m 2: 4, as Rodl and Sidorenko [35] showed. The example of Rodl and Sidorenko is based on the usual Lagrangian function of a multigraph. Given a multigraph G and a vector x E ~ (IGI) we define the quantity A(G,X) = L:uvEE(G)XUx v' The Lagrangian of G is then A(G) = max { A(G, x ) : x E ~}. The central element of their

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argument is a (wonderfully elegant) proof that, for each integer d ~ 3 and each integer k, there exists a multigraph F, in which each pair of vertices is joined by either d or d + 1 edges, such that A(F) > d, but for each submultigraph F' c F with IF'I ::; k, A(F') < d holds. (This multigraph is the one in Theorem 2.4 of [35], its properties being given in Corollary 2.3 and the proof of Theorem 2.7.)

Theorem 3.28. Let l ~ 5 be an integer and let p = 11l. Then there exists a family H of 2-coloured multigraphs for which there is no finite sub-family 1io c 1i with K,p(1io) = K,p(1i). Proof. For each complete 2-coloured multigraph H we define the (ordinary, uncoloured) multigraph H m to be the multigraph with vertex set V (H), such that two vertices u, v E V(H) are joined in H m by 0, l - 2 or I - 1 edges according as the edge uv is red, blue or green . It is important to the argument that if H c H' then H m C H:n (though the converse does not hold). Observe that if H is a complete 2-coloured multigraph and x E ~(IHI) is the vector with all co-ordinates equal to 1I IHI, then A(H m- x) equals 2(l- 2)/IHI 2 times the number of blue edges in H plus 2(l- 1)/IHI 2 times the number of green edges. Since p = 11l we have wp(H) = p(I~I) + IHI 2 A(H ,x )21 ' m

Let 1i = {H : A(Hm ) > l - 2}. By the definition of K,p(1i) there is a sequence (Gn) of complete 2-coloured multigraphs, none of which contains any member of 1i, with IGnl = n and with wp(Gn) = (K,p(1i) + 0(1)) (~). But wp(Gn)::; (P+A(G~)ll+o(l)) (~). Since, certainly, G" 1: H, we have A(G~) ::; l- 2. Thus K,p(1i) ::; (l- l)ll = q. Suppose now that 1io C 1i and 1io is finite. Let k = max {IHI : H E 1io}. Let F be the multigraph of Rodl and Sidorenko defined above, taking d = l- 2. Then F = Gm for some complete 2-coloured multigraph G, none of whose edges are red. Let T be the type obtained by colouring each vertex of G red. We claim that no H E 1io is r-colourable. For suppose instead that Hem x T . Since IHI ::; k, there is a sub-type (J C T , with I(JI ::; k, such that Hem x (J. As noted at the beginning of the proof, this means Hm C (m x (J)m' which in turn implies A(Hm) ::; A( (m x (J)m)' The vertices of (J span a sub-multigraph F' C F with IF'I ::; k, so A(F') < l - 2 by the property of F. But (m x (J)m is obtained from F' by replacing each vertex of F' by a class of m vertices, with vertices in different classes

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being joined to each other by edges of multiplicity equal to the multiplicity between the original vertices of F'. It is a standard and simple property of the Lagrangian that A( (m x o-)m) = A(F'). So A(Hm) < l-2, contradicting

HE 1£0. Because no H E 1£0 is r-colourablo, Observation 3.7 implies /'i,p(1£o) ~ Ap(T). Let x E ~(ITI). Since p = l/l, and by the construction of Gm = F, we have XtWp(T)X = p(l+A(F,x)) . Thus Ap(T) = p(l+A(F)) > p(l-l) = q. Hence /'i,p(1£o) > /'i,p(1£), completing the proof. •

Remark 3.29. It is interesting to note that Theorem 3.28 fails for p = 1/2; in this case, there is always a suitable finite 1£0 C 1£. This is because the only p-core types with p = 1/2 are the types T(a, b) , comprising a red and b blue vertices joined to each other by green edges (see Example 5.2), and , for any class 1£, /'i,1/2(1£) = 1 - 1/2t where t is the maximum value of a + b for which there is a T(a, b) , such that no H E 1£ is T(a, b)-colourable. We can, for each j with 0 ~ j ~ t + 1, choose some Hj E 1£ which is T(t + 1 - j ,j)-colourable. Then, putting 1£0 = {H o, ... , Ht +1 } we obtain a finite sub-family 1£0 C 1£ with /'i,1/2(1£0) = /'i,1/2(1£).

3.6. Incomplete extremal graphs As promised at the start of §1, we now look briefly at what can be said about the extremal function for 2-coloured multigraphs if we do not constrain the extremal graphs to be complete. That is, we study the function

eXp(H, n) = max {wp(G) :

IGI = n,

rather than kexp(H, n), and we define J.Lp(H)

H

et G}

= limn -+ oo eXp(H, n) (~r1 .

In many small cases where /'i,p(H) and J.Lp(H) can be evaluated, they are equal , though this is not always the case (see Example 5.10) and there is no immediate reason to think that equality is common. However if, say, H has only blue and green edges, then /'i,p(H) = J.Lp(H), because the addition of red edges to an extremal graph which is not complete will not create a copy of H. To extend the methods of §3 to accommodate incomplete extremal graphs, we need to extend the notion of a type to allow white vertices and edges, which have weight zero. If T is such an extended type then the

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definition of m x r would preclude any edges within classes corresponding to white vertices or between two classes corresponding to a white edge. The results of sections §3.1 and §3.2 would then still hold good in this extended context, because of their generic nature; we define Ap(r) for an extended type in the same way as for an ordinary type, though of course the p-matrix Wp(r) might contain some zero entries. For the same reason the initial simple results in §3.3 would still hold, such as Facts 3.19 and 3.20, and of course Proposition 3.21. After that, things start to go wrong. Lemma 3.22 still holds good as stated, but the proof does not extend to showing that the edges at each vertex of a p-core type have weight greater than that of the vertex. So Theorem 3.23 is weakened to stating that (if p > 1/2) all edges meeting a blue vertex are green, together with whatever information is supplied by Fact 3.20. The main casualty is, of course, Theorem 3.25; the possibility of non-blue edges between non-blue vertices destroys the heart of the proof. The counterexamples of §3.5 still stand. Theorem 5.6 still gives infinitely many graphs with the same value of Ap and, because J1p(H) = Kp(H) for the graph H of Theorem 3.27 (see [31]), the theorem remains true. Theorem 3.28 goes through essentially without change: the graphs H m are defined in the same way except that white edges, like red edges, contribute nothing to H m , and the equation given for wp(H) remains true as an upper bound, which is all that is needed. An application of the parameter J1p is discussed in the next section.

4.

GRAPH GAMES AND INCOMPLETE EXTREMAL GRAPHS

There is a rich and fascinating connection between games on graphs and Ramsey theory, described in detail by Beck [9] . One such game, played on the edges of a complete graph K n by two players, was studied by Richer [34]. Before the game, some 2-coloured complete graph H is specified, which has no green edges. The first player, when it is his turn, selects some fixed number of edges of K n and colours them red or blue. The second player, in turn, selects an edge and labels it +. The first player wins if, at the end of the game (when all of K n is coloured), there is a copy of H in K n all of whose edges are labelled +. Whether the second player can win depends on how many edges the first player must colour at each turn.

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A simpler version of this game is one where the first player selects outright a red/blue colouring of K n and the second player then labels some prescribed number of the edges. The first player wins if the second player cannot avoid a copy of H within the labelled edges. In this form of the game, the parameter of interest is f(H), which is the minimum value of f such that the first player can win if n is large and the second player must label f(~) edges. More generally, let fp(H, n) be the minimum number of edges the second player must be made to label in order that the first player can find a winni~? colouring with IP(~)l red edges, and let fp(H) = limn-toofp(H,n)G) (the existence of this limit is shown by the proof of the next theorem). Then we can define f(H) = min, fp(H). It turns out that f( H) behaves similarly to the parameter ed (Forb (H) ) of §2.2 except that, rather than /'\,p(H) , it is the "incomplete" extremal function f.lp(H) that is involved (see §3.6).

Theorem 4.1. For the game above, fp(H) = f.lp(H) and so f(H) min; f.lp(H). Proof. Rather than dot every 'i' and cross every 't' we shall sketch the proof in the spirit of §2.2; however, unlike in the case of edit distance, the application of Szemeredi's Lemma is standard, and the reader familiar with this Lemma and with standard facts about random graphs, such as are frequently used in the works on hereditary property and edit distance cited in this paper, can easily fill in the details if minded to do so. We suppose that n is large, that the first player must colour p(~) edges red, and that the second player must label f G) edges. If f < J-Lp(H) , the second player chooses a 2-coloured graph F of order n containing no H and with wp(F) 2: fG). Remember that the underlying graph of F need not be complete , but the missing edges just have weight zero. Now, whatever 2-coloured K n is chosen by the first player, the second player finds a bijection of V(F) to V(Kn ) which maximizes the number of edges of K n agreeing in colour with the corresponding edge of F; a green edge of F will agree with an edge of K n of either colour, whereas a white (absent) edge of F agrees with neither colour. Since K n has p(~) red edges, any given red edge of F has probability p of agreeing with its corresponding edge in K n , and so the expected number of agreeing edges in a random bijection is wp(F). Hence by labelling the agreeing edges in a bijection of maximum agreement, the second player labels at least £ (~) edges, and

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these all lie in F so contain no copy of H . Hence the second player wins if £ < j1p(H), and so £p(H, n) ~ j1p(H) G). On the other hand, if £ > j1p(H) + 6c:, then the first player selects a random colouring of K n with red probability p. The second player then labels £(~) edges; let L; and L b be the red and blue labelled subgraphs. Find a partition of V(Kn ) that is a Szemeredi partition simultaneously for both L; and Li; with k > 2/E classes and 'TJ = 'TJ(H, E)-regular pairs (this exists, see [29, p. 306]). Form the 2-coloured reduced graph R of order k, whose red edges represent regular pairs of density > 'TJ in L; and whose blue edges behave similarly with respect to Lb. The number of labelled edges is at most the edges within classes plus the edges in irregular pairs plus the edges in low density pairs plus the edges in regular pairs . Since the colouring is random, almost surely there are at most (p + E)(n/k)2 red edges between any pair of n/k sets in the colouring, and so in particular at most this many edges of L: in a pair corresponding to a red edge of R. Hence the number oflabelled edges is at most 4c:G) + wp (R)(n/ k)2. Thus

wp(R) ~ (j1p(H) + c:)(~) ~ ex (H, k) if k was chosen large enough, and so R contains a copy of H . By the standard property of Szemeredi partitions, H can now be constructed in L; U Li, and the first player wins. Hence £p(H, n) :::; (j1p(H) + 6E) (~) for large n . _

The proof shows that random colourings are always a good choice for the first player, though there are sometimes other choices possible. The fact that random colourings are good is made even clearer in the more general game studied by Richer, for which some number 0 :::; t :::; (I~I) is also specified before the game, and the first player wins if the second player is forced to label at least t of the edges of some copy of H . (So up till now we have been playing this game with t = (I~I) .) Let 11. be the class of graphs on V (H) comprising t edges of H . Then the first player wins if the second player must label at least min; j1p(lI.) edges and otherwise the second player wins. The above proof works for the general game without change until the very end , where we need to observe that R contains some graph H' comprising t edges of H . As before, the pairs corresponding to edges of H' are regular in L; or Lb as needed. But the other pairs correponding to edges of H \ H' are regular in the red or blue edges of K n because the colouring is random. So we can construct a copy of H in K n in which t edges, the edges in H', are labelled . In [34] Richer gives many examples and much information about £(H) and £(11.) for the corresponding families 11., and in particular all values for

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IHI :S 4. Exact values of ex (H, n) are given for all n and all H with IHI :S 3 and no isolated vertices; there are ten such graphs. His work necessarily contains values for J1p(H) in many instances. For example, let H be the 2coloured complete graph of order 4 having one green edge, a disjoint red edge and a blue K 2 ,2 in between. Then Kp(H) = J1p(H) = max {1- q/2, 1- pq}, the extreme types being either two red edges joined by a green edge, or one blue and one red vertex joined by a green edge. The most noteworthy question left open by Richer is the following conjecture, having the flavour of the Erdos-Storie theorem. Conjecture 4.2 (Richer [34]). Let H be a 2-coloured multigraph with no green edges. Then there exists some value of p such that 1

Kp(H) :S J1p(H) :S 1- IHI _ 1

if

IHI is odd.

Notice that the conjecture is best possible, insofar as if H is complete then a complete (IHI - 1) -partite graph coloured green contains no copy of H . This colouring corresponds to the extended type with IHI- 1 white vertices joined to each other by green edges. The conjecture implies Kp(H) :S J1p(H) :S 1- 1/IHI for IHI even. The stronger inequality Kp(H) :S J1p(H) :S 1- 1/(IHI- 1) does not always hold when IHI is even; for example , if H; = Kt ,t and Hi, = H r = 2Kt then min, Kp(H) = 3/4 when t = 2 and min; Kp(H) > 1 - 1/(2t -1) when t = 3 and t = 4 (see §5.3). However the stronger inequality might hold when IHI is even and large, perhaps if IHI > 8. There is a cognate conjecture of Diwan and Mubayi [20, Conjecture 3] which is interesting in this respect. It asserts that , if H is a 2-coloured multigraph with no green edges, then a 2-coloured multigraph G on n vertices with min { e( G r ) , e( Gb)} > (1-1/ (IHI-l) )n2 /2 will contain H unless IHI = 2t with t = 2, 3 or 4, H is complete, and the edges of one colour form a spanning subgraph whose components are all balanced complete bipartite graphs. Richer proved his conjecture for IHI :S 5, and established the general bound minpJ1p(H) :S 1- 3/(5IHI- 5). Further results are given in [31].

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E. Marchant and A . Thomason

EXAMPLES

In this section we discuss the value of Kp(H) for various graphs H . To find Kp(H) we find a p-core type r whose existence is guaranteed by Theorem 3.25, such that H is not .r-colourable and Ap(r) = Kp(H) . We divide the examples into two classes, those in which the edges of r are all green, which we call the simple examples, and those in which red or blue edges can appear. Theorem 3.23 helps to reduce the number of types that need be considered , but even with this theorem there can remain significant difficulties. We begin by giving names to those important core types which contain only green edges. Definition 5.1. The type r(a, b) has a red vertices and b blue vertices, joined by green edges. Thus Q( r( a, b)) consists of all complete 2-coloured multigraphs whose vertices can be split into a + b classes, a of which span only red edges and the other b spanning only blue edges. These are the 2-coloured analogues of the hereditary properties P(a + b, a) found in [11] . The vector x E ~ which assigns vertex weights in the proportions p :: q to red and blue vertices is optimal by Fact 3.19. It follows that Ap ( r(a, b)) = 1- pq/(ap + bq) , and that r is p-core if p =1= 0,1. We shall generally assume that p =1= 0, 1 since these extreme cases are covered in §3.1. On occasion we might also assume that p =1= 1/2, since Kl/2(1-l) is treated in Example 5.2 for all n. 5.1. Simple examples

Example 5.2. Suppose p = 1/2. Then , by Theorem 3.23, p-core types contain only green edges, so the types r(a, b) are the only such types. Now Al/2( r(a, b)) = 1-1/2(a+b). Therefore, for any class 1-l, Kl/2(1-l) = 1-1/2t where t is the maximum value of a + b for which there is a r( a, b), such that no H E H is r(a, b)-colourable. (This value of "'1/2(1-l) appears also in [5] . The number t is the colouring number mentioned after Definition 3.3; t+ 1 is the binary chromatic number studied in [7], where this result is also given.)

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Example 5.3. Let r be a p-core type and let 1i(r) be all graphs that are not r-oolourablo; that is, 1i(r) is exactly those graphs not in Q(r) . So Ap(r) ::; K,p(1i(r)) by Observation 3.7. On the other hand, any sequence of extremal graphs for the class 1i(r) must , by definition, lie in Q(r) and so K,p(1i(r)) ::; Ap(r) by Remark 3.6 and , indeed, f.Lp(1i(r)) ::; Ap(r) for the same reason . Thus K,p(1i(r)) = f.Lp(1i(r)) = Ap(r) for all p.

In particular, K,p(1i(r(a,b))) = f.Lp(1i(r(a,b))) = 1 - pql(ap + bq). Note that K,p (1i (r(a, b))) is minimized when p = Vbl( + Vb), and then K,p(1i( r(a ,b))) = 1 - II (

1/(

va + Vb)

2,

va + Vb)

va

2

or, equivalently, ed (P(a, b))

as proved in [5] .

The following bound will prove useful for some more elementary examples. Lemma 5.4. Let r be a p-cote type whose vertices are all red, each joined to at most d others by green edges. Then Ap(r) ::; max { q, 1 - ql(d + I)}. Proof. Let the maximum number of green edges meeting any vertex be m::; d. Then Irl ~ m + 1. Let x E .6.( Irl) be a vector with Ap(r) = xtWp(r)x. Let G be the graph with vertex set V(r) whose edges are the green edges of r and let d(u) denote the degree of the vertex u in G. By Fact 3.19 we have Ap(r) = pXu+(I- x u)q+p L:uvEE(G) Xv for every u E V(r) . Summing this equation over all u we obtain IrIAp(r) = p - q + qlrl + p L:u d(u)x u ::; p - q + qlrl + pm. Hence Ap(r) ::; q + (p(m + 1) - q) Ilrl. If p(m + 1) - q ::; a we have Ap(r) ::; q; if p(m + 1) - q > 0, we have Ap(r) ::; q + (p(m + 1) - q) I(m + 1) = 1 - ql(m + 1) ::; 1 - ql(d + 1). • Example 5.5. Let H; = K a + K band Hi, = H r = Ki; that is, IHI = a + b, H is complete with no green edges, and consists of a red K a joined to a blue Ki, by red edges. In particular, if a = 1, the red graph H; is a star K1,IHI-l' Note that H contains a blue Ki, and a red K a+1 , so is neither r(b - 1, O)-colourable nor r(O, a)-colourable. Hence K,p(H) ~ max {I - pia, 1 - ql(b -I)}. Let r be such that H is not r -colourable and Ap(r) = K,p (H) , as given by Theorem 3.25. Then r cannot have both a red vertex and a blue vertex for, if so, they would be joined by a green edge by Theorem 3.23, and H would be .r-colourable. So the vertices of r are all the same colour. If they are all red, the edges of r are green and blue, so no vertex can be joined to b - 1 others by green (else H is r-colourable) ; by Lemma 5.4, this means

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Ap (7) ::; 1 - ql(b - 1). If the vertices of 7 are all blue, they are joined to each other by red and green, and none can be joined to a others by green (else H is r-colourable} , so Ap ( 7) ::; I-pia by the complementary version of Lemma 5.4. Thus we have shown that /'\.p(H) = max {I-pia, l-ql(b-l)}. The edit distance of H with (a,b) = (1,3) was computed in [5] as an example where the minimum value of p is not 1/2 (in this case it is 1/3). The edit distance of H for all (a,b) was given in [8]. A more precise estimate of kex(H, n) is given in [31]. As discussed in §3.5, the graphs of Example 5.5 can sometimes have infinitely many 7 satisfying Theorem 3.25. For example, if (a, b) = (1,4), that is, H; is the star K 1,4, then /'\.1/4(H) = 314, and the next theorem shows that any type whose vertices are red and whose green edges comprise a spanning cycle will do.

Theorem 5.6. Let d ~ 2 be an integer and let G be a connected d-regular graph of order n. Let 7G be the type with vertex set V(G) whose vertices are all red, with the edges of G coloured green and all other edges blue. If p::; I/(d + 2) then

Proof. Let W Then

7G

is p-core, and Ap (7G) = q - (1- p(d + 2)) In.

= Wp (7) and let x

xtwx=q(

L

xv

E

8.. Let A be the adjacency matrix of G.

)2-(q-p) L x~+pxtAx

VEV(T)

=q-(q-p-pd)

VEV(T)

L

X~-P(d

VEV(T)

=q-(I-p(d+2))

L VEV(T)

L

X~-xtAX)

VEV(T)

x~-p

L

(x u - xv )2.

UVEE(G)

Since G is connected, both sums of squares are minimized only when Xv = lin for all v E V(7) . Because 1 - p(d + 2) ~ 0, the quantity xtWx has a unique maximum at this point. • The preceding theorem does not preclude the type 7G being p-core for values of p larger than 1I (d + 2). For example, if d = 2 and G is an n-cycle, then Sidorenko's criterion (Proposition 3.21) can be called on to show that 7G is p-core if and only if p < II (2 + 2 cos (21f In)) ; we omit the calculation.

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273

Example 5.7. Let H; = K1,b and Hi, = Kb+l; thus IHI = b + 1 and H consists of a green star K l ,b with the remaining edges blue. As remarked in §3.6, in this case, /'l,p(H) = J-tp(H) because missing edges in the extremal graphs can always be replaced by red edges. We observe that H is neither r(O, l)-colourable nor r(b, O)-colourable, and so /'l,p(H) 2: max {q, 1 - q/b}. Let r be an optimal p-core type as in Theorem 3.25. Suppose p > 1/2. Any blue vertex of r can be replaced by a red vertex without H becoming rcolourable, and the p-value of r has been increased. Since r was optimal, it must be that the vertices were already red. So r must have only red vertices, and by Theorem 3.23 these are joined by green edges. There cannot now be b + 1 vertices, so r C r(b, 0) and /'l,p(H) ::; 1 - q/b. Suppose instead p < 1/2. If r has a blue vertex v then it is joined to the others by green, by Theorem 3.23; if there is another then H is r-colourable, so r=r(O, 1). Finally, if all the vertices are red, then these are joined to each other by blue and green edges, so no vertex can be joined to more than b-1 others by green. By Lemma 5.4, /'l,p(H) ::; 1 - q/b. We conclude that /'l,p(H) = max {q, 1- q/b}. A more general result with H;

= K a + Kb and Hi, = H; is proved in [31] .

Example 5.8. Let H; be a star with perhaps an extra pendant edge added at one of its leaves, together with perhaps some isolated vertices. Let Hi, = Hr . Let s = 2 if H; has an extra pendant edge and s = 1 otherwise; thus H; contains s disjoint edges. Hence X(Hb) 2: IHI - s and so H is not r(IHI- s -l,O)-colourable. Certainly H is not r(O , l)-colourable because H; is non-empty. Thus /'l,p(H) 2: max {q, 1- q/(IHI- s -1)}. On the other hand , H is r(O, 2)-colourable because H; is bipartite. His also r(l , l)-colourable: to see that Hem x r(l, 1) for large m, place the centre of the star plus its neighbour with the pendant edge (or any leaf if s = 1) in the red class and place all other vertices in the blue class. Let r be a p-core type for H with /'l,p(H) = Ap(r). Suppose r contains a blue vertex. If Irl = 1 then Ap(r) = q. If Irl > 1 then, because neither r(O , 2) C r nor r(l, 1) C r, there can be no green edge meeting a blue vertex. But then there can be no red vertices (by Theorem 3.23) and since the edges are red it must be that p > 1/2. Thus Ap(r) ::; p::; 1- q/(IHI- s -1). On the other hand, if all vertices of r are red, then they are joined by green and blue edges. If some vertex u E V (r) is joined to IH 1-s - 1 others by green, then H is contained in m x r: place the star centre plus a leaf in the class Vu , if s = 2 place the neighbour with the pendant vertex together with

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its pendant edge in some class Vu' where uu' is green, and place all other vertices in distinct classes Vv where uv is green. So no vertex of 7 is joined to IHI- s -1 others by green and so Ap(7) ::; max{q, 1- q/(IHI- s -I)} by Lemma 5.4. In summary, Kp(H) = max{q,l- q/(\H\- 2)} if H; is a star plus isolated vertices, and Kp(H) = max {q, 1- q/ (IHI- 3)} if H; is a star with an extra pendant edge plus isolated vertices. Example 5.9. We can now easily write down Kp(H) for all H with Hb = H r and IHI ::; 4. The edit distance for these graphs was computed in [7] for IHI ::; 3, and for IHI = 4 in [5], with an alternative proof in [8]. If H is monochromatic then Kp(H) is given by Turan's theorem; for red H we have Kp(H) = q+p(I-I/(IHI-l)) = 1- p/(IHI-l). In the other case when IHI = 3, there is a single edge in one colour, say red; this case is covered by Example 5.8, and Kp(H) = max{q ,p} . Four other cases with IHI = 4 are covered by the same example; when the red edges comprise a star then Kp(H) = max {q, 1 - q/2} and when H; is a path of length three then Kp(H) = max {q,p} .

The final case is when H; = K 2 ,2 and Hi, = H; = 2K2 . Since H is not 7(1,1)-colourable we have Kp(H) ~ 1 - pq. Let 7 be a p-core type for H as in Theorem 3.25. Then 7 cannot have two blue vertices, since thes e must be joined by red or green and H would be .r-colourable. But H cannot have just red vertices, since either two are joined by green, in which case H is 7-colourable , or all are joined by blue, in which case Ap(H) ::; max {p,q} < 1 - pq. Thus 7 has exactly one blue vertex , joined by green edges to the other (red) vertices. Now there cannot be two red vertices else these are joined by blue or green and H is .r-colourable. So 7 = 7(1 , 1) and Kp(H) = 1 - pq. For further examples , such as paths and cycles, and for more precise values of kex(H, n) in some instances, the reader is referred to [31].

5.2. Less simple examples In the examples of §5.1, some type 7(a, b) whose edges are entirely green is always optimal. It is not immediately clear that there are graphs H which are not of this simple kind . The first example was provided by Balogh and Martin [8], who displayed a graph H9 of order 9 for which the edit distance

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1- minpl'l:p{H) is not realized by a type 7{a,b). We give a slightly smaller example which can be analysed fully.

Example 5.10. Let H; be a six-cycle with a diagonal and let Hi, = Hr. We denote this graph H by C6. Let a be the type with one blue and two red vertices, the red vertices being joined by a blue edge and the other edges being green. The vector assigning weight 1/{1 + 2p) to the blue vertex and p/{1 + 2p) to each of the others shows Ap{cr) 2:: 1 - p/{1 + 2p). The only subtype of o with p-value greater than 1 - min {p, q} is 7{1, 1) with p-value 1 - pq. Therefore o is pcore for < P < 1/2, and moreover the vector just described is optimal by the converse of Fact 3.19 stated after Fact 3.20, so Ap{cr) = 1 - p/{1 + 2p) .

°

Certainly C6 is not 7(2, O)-colourable, because Hi, is not bipartite. Consider a possible embedding of C6 in m x CT . If we place a triangle of Hi, in the blue class, there is a disjoint blue triangle which has to be placed amongst the two red classes, which is impossible. So at most two vertices can be placed in the blue class and, since no more than two can be placed in a red class, there must be exactly two in each class. The four vertices in the two red classes must comprise two red edges joined by a blue K 2,2 . There is only one such subgraph in C6, and the remaining two vertices span a red edge, which cannot be placed in the blue class. Therefore C6 is not o-colourable. Hence I'I:p ( C6) 2:: max { 1 - q/2, 1 - p/ (1 + 2p)} . Now let 7 be an optimal type for C6as given by Theorem 3.25. Observe that Cli is 7{0, 2)-colourable because H; is bipartite. Moreover Cli has three disjoint edges coloured red, red, blue and so it is 7(2, l l-cclourable. Notice too that C6 contains four vertices spanning a blue K 2 ,2 plus two red edges, with the other two vertices joined by red; thus 7 cannot contain a red vertex joined by green to two other red vertices (these two must be joined by blue or green). Suppose first that p < 1/2, so the edges of 7 are green and blue by Theorem 3.23. Since 7{0,2) ct 7, there is at most one blue vertex. If 7 has a blue vertex it is joined by green to the red vertices, and there can be at most two of these (else C6 would be r-colourable because red vertices are joined by blue or green) . Because 7{2,1) ct 7, if there are two red vertices they must be joined by blue; thus 7 c cr. Suppose on the other hand that there are no blue vertices. We know that a red vertex can be joined by green to at most one other red vertex. Then Lemma 5.4 shows that "'p(7) S max {q, 1 - q12} < 1 - pI(l + 2p).

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Suppose now that p ;::: 1/2. The edges of 7 are green or red. Since 06 is both 7(2 , l)-colourable and 7(3 ,0)-colourable (because H b is 3-chromatic), if 7 has two red vertices it can have no other vertices so 7 = 7(2,0) . If 7 has only blue vertices, they must be joined by red edges because 7(0,2) ct 7, but then Ap (7) ~ p ~ 1 - q/2. Thus 7 has exactly one red vertex. There must be at least two blue vertices else Ap (7) ~ Ap ( 7(1,1)) = 1-pq ~ 1-q/2. So 7 contains a red vertex joined to two blue vertices by green, and the blue vertices are joined by a red edge. But 06 contains four vertices spanning a red K2,2 plus two blue edges, with the other two vertices joined by red; so in this case 06 is .r-colourable. We conclude that K p (C6) = max {1 - q/2, 1 - p/(l + 2p)} , and that when p =1= 0, ~ ,1 then a and 7(2,0) are the only optimal types. As noted in §3.6, 06 gives an example of a graph for which K p =1= /lp, because a complete 5-partite graph coloured green does not contain 6, and therefore /lp( 06) ;::: 4/5 for all p. Hence /lp( C6) > K p(C6) for 1/3 < p < 3/5. The next example requires less work and will be helpful in §5.4. It is based on the following lemma about green triangles, which is itself straightforward and useful.

°

Lemma 5.11. Let 7 be a type whose vertices are all red and which contains no green triangle . If p ~ 1/2 then Ap (7) ~ (p - q)/171 + q + p/2 ~ 1 - p/2. Proof. Consider the graph of order n in Q(7) of maximum weight, as in Remark 3.6. It has no green triangle so by Turan's theorem it has at most n 2/4 green edges. So if x is a vector maximizing Xt W p (7)X then Ap(7)G) = wp(G) +O(n) ~ ((p- q) l:u x~ +q+p/2) (~) +O(n) , and hence Ap (7) ~ (p - q) l:ux~ + q + p/2. Since l:u x~ ;::: 1/171 and p - q < 0 we obtain Ap (7) ~ (p - q)/171 + q + p/2 . • Example 5.12. Let HI be the graph of Example 5.7, namely, a green star KI ,b with the remaining edges blue. Let H 2 be a green K 3 and let 1£ = {HI , H 2} . Neither graph in 1£ is 7(0,1)-colourable or 7(2,0)-colourable. Nor is either o-colourable, where a is the type of order 2(b - 1) with red vertices, whose green edges form Kb-I ,b-I and whose remaining edges are blue. The uniform weighting shows that Ap(U) ~ q + p/2 + (p - q)/2(b -1) . Therefore K p (1i ) ~ max { q, 1 - q/2, q + p/2 + (p - q)/2(b - 1)} . Let 7 be an optimal p-core type for 1£. If p > 1/2 then any blue vertex of 7 can be re-coloured red without HI or H2 becoming .r-colourable, and this would increase Ap (7). Therefore all vertices of 7 must be red and joined

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by green edges, so 7 C 7(2,0) . Thus K,p(Ji) ~ 1 - q/2, and the same holds for p = 1/2 by Theorem 3.1. So suppose p < 1/2. If 7 has a blue vertex v then, as in Example 5.7, there can be no other vertex, so 7=7(0,1) and Ap (7) = q. This leaves the case where 7 has only red vertices, joined by blue or green edges. As in the proof of Lemma 5.4, the equation Ap ( 7) ~ q + (p(m + 1) - q) /171 holds , where m is the maximum number of green edges meeting a vertex. Clearly m ~ b - 1 because HI is not r-colourable. If p(m + 1) - q < 0 we have Ap (7) ~ q, and otherwise, if 171 2: 2(b - 1), we have Ap (7 ) ~ q + (p(m + 1) - q) /2(b -1) ~ q + p/2 + (p - q)/2(b - 1). By Lemma 5.11 this inequality still holds if 171 ~ 2(b-1) , bearing in mind that p- q < O. We conclude that K,p(Ji) = max {q , 1- q/2, q + p/2 + (p - q)/2(b -I)}. Remark 5.13. Example 5.12 shows that the strict inequality K,p ( {HI, H 2 } ) < min {K,p(Ht},K,p(H2)} is possible, because K,p(Ht} = max{q, 1- q/b} by Example 5.7, and K,p(H2) = 1/2+max {p/2 , q/2} by Turan's theorem. Thus the inequality is strict for l/(b + 1) < p < 1/2.

5.3. Complete bipartite examples In what remains, we study H with H; complete bipartite and Hi, = Hr . Examples 5.5 and 5.9 show that the cases H; = KI ,t and H; = K 2 ,2 can be treated easily, but even H; = K 2 ,t becomes much more involved. We first prove a general fact about optimal types for these graphs. Lemma 5.14. Let H; = Ks,t where 8 ~ t and let Hi, = Hr . Let 7 be a p-core type such that H is not r -coioureble and K,p(H) = Ap(7 ), as guaranteed by Theorem 3.25. Then either 7 = 7(8-1, 1) and K,p(H) = 1-pq/(1+p(8-2)) , or all the vertices of 7 are red .

If p > ~, then 7 = 7(8 - 1,1) if t = 8 and 7 = 7(t - 1,0) otherwise. Proof. We use Theorem 3.23 repeatedly. There cannot be two blue vertices in 7, for they would be joined by a red or a green edge and then H would be r-oolourable. If 7 has exactly one blue vertex, then it is joined to the other vertices by green edges. The red vertices are joined to each other by blue or green edges, so if there are 8 of them then H would be r-colourable (by placing t vertices in the blue class of m x 7 and the other 8 vertices in distinct red classes). Thus T has at most 8 - 1 red vertices and so 7 C 7(8 - 1,1).

278 Since H is not

E. Marchant and A. Thomason 7(S -

1, 1)-colourable, it must be that 7 =

7(S -

1,1) and

K,p(H) = Ap(7) = 1- pq/(1 + p(s - 2)). If p > ~ and 7 f= 7(S - 1,1) then all the vertices are red and all the edges are green. Since H is 7(t, O)-colourable but not 7(t -1, O)-colourable, it must be that 7 = 7(t - 1,0). Now Ap ( 7(t - 1,0)) = 1 - q/(t - 1), which is larger than Ap ( 7(S -1,1)) if p > 1/(t - S + 1). •

Given an ordinary uncoloured graph G, let 7C be the type with vertex set V(G) whose vertices are all red, with the edges of G coloured green and all other edges blue. By taking the vector x = (1/IGI, . .. , 1/IGI) we have

where e(G) is the number of edges of G and d is the average degree. By Lemma 5.14, the determination of K,p(H) comes down to finding those G with Ap ( 7C) > 1 - pq/ (1 + p( S - 2)) for which H is not 7c-colourable, if indeed there are any. But this task is not straightforward. Remark 5.15. If His 7c-colourable, that is, Hem x 7C for some m, then the vertices of the blue K; lie in different classes, as do the vertices of the blue Ki . So there is some number c, 0 ~ c ~ S of classes, which contain both a vertex of K; and a vertex of Ki, and the remaining s + t - 2c vertices are in distinct classes. The c classes that contain two vertices must be joined by green to all the other s + t - c - 1 other classes. This implies that G contains K; + Ks-c,t- c as a subgraph. Notice that Ks-l + Kl,t-s+l contains K s + Ko ,t-s = K; + K t- s, and if s = t then so does K S - 2 + K 2,t-s+2. Therefore, the condition that H be not 7c-colourable is that none of K; + K t - s and K; + Ks-c,t-c, 0 ~ C ~ S - 2, are subgraphs of G, though the case c = s - 2 is redundant if s = t. Example 5.16. Let H; = K2,t and Hi, = Hr. We know from Example 5.9 that 7(1,1) is optimal for all p if t = 2, and K,p(H) = Ap( 7(1,1)) = 1 - pq = q + p2. However, this is not true for any t ~ 4. If G is a graph containing neither K 2,t nor K 2 + K t-2 then H is not 7c-colourable, by Remark 5.15. Take such a graph G which is large and reasonably dense; then 7C will be better than 7(1,1) for some small values of p. In fact, the graph G = K 3 ,3 already gives some information. We have IGI = 6 and d = 3 so Ap (7C ) ~ q + (5p -1)/6> q + p2 if 1/3 < p < 1/2. So, if t ~ 4,7(1,1) is non-optimal for 1/3 < p < 1/2.

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Example 5.17. Let H; = Ks,s and H b = H r, where S ~ 2. Let G = K 2s- 2 - (s -1)K2, that is, G is a Ks-1 ,s-1 with a l-factor removed. Then G doesn't contain K; or Kc+Ks-c,s-c for any c, 0::; c::; s-3, and so H is not 7c-colourable. We have IGI = 2s-2 and d = 2s-4 so Ap (7C) ~ 1-1/(2s-2) . Hence Ap (7C) > Ap ( 7(S -1,1)) if p2(2s - 2) - sp + 1 < 0. Since the roots of this quadratic lie between zero and one, there is a value of p for which the inequality holds provided s2 - 8(s - 1) > 0, or s ~ 7.

The meaning of this for edit distance (see §2.2) is that, for all s ~ 7, the edit distance for Forb (Ks,s), namely 1 - min, Kp(H), is realized by a type 7 not of the form 7(a, b), because by Lemma 5.14 a type with only green edges would have to be either 7(S - 1,1) or 7(S - 1,0). Clearly Ap (7(S -1,0)) ::; Ap (7(S - 1,1)) for all p, and the minimum value of Ap ( 7(S - 1,1)) is less than 1-1/(2s - 2) = Ap (7C)' Another choice of G which works here, and which is a slightly better choice, is G = K2s-1 - (s - 2)K2 - P3, where P3 is a path of length two disjoint from the missing edges.

5.4. K3,3

The case H; = K 3 ,3 and Hi, = H r is a particularly interesting one. It appears as a special example in the conjectures of Richer (Conjecture 4.2) and of Diwan and Mubayi (in §4), and it was mentioned by Alon and Stav [5] as the smallest outstanding case whose edit distance they had not computed. The edit distance was subsequently shown by Balogh and Martin [8] to be 3 - 2V2, this being 1 - min, Ap ( 7(2, 1)) , minimized at p = V2 - 1. As shown by Lemma 5.14, Kp(H) = Ap( 7(2,1)) for p > 1/2, and this raises the question whether this equation holds for all p. In fact, it is relatively easy to show that it holds for p > 1/3 (Example 5.18), which is enough to evaluate the edit distance, so giving an alternative and somewhat simpler proof than that in [8]. It turns out that the equation does not hold for small p (Example 5.19), though the argument is a little delicate. After giving these examples we shall examine for which values of p the equation can be shown to still hold. Example 5.18. Let H; = K 3 ,3 and Hi, = Hr. Let 1/3 < p < 1/2 and let 7 be a p-core type for which H is not .r-colourable and with Kp(H) = Ap(7), as in Theorem 3.25. Then Ap (7) ~ Ap ( 7(2,1)) = 1- pq/(l + p) > 1- p/2. Lemma 5.14 shows that either 7 = 7(2,1) or all the vertices of 7 are red,

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but the latter is impossible by Lemma 5.11, for then 7 would contain 7(3,0) and H is 7(3, O)-colourable. The minimum of Ap( 7(2,1)) occurs at p = V2 -1> 1/3, and so Kp(H) has a local minimum at this point. By Theorem 3.1 this local minimum is also a global minimum, and so the edit distance of Ka,a is 3 - 2V2. Example 5.19. Let H; = Ka,a and Hi, = Hr. Let G be a graph that contains neither K a nor Ka,a. Then H is not 7c-colourable by Remark 5.15. As pointed out earlier, Ap (7C) ~ q+ (p(d+2) -1) /IGI, whereas Ap( 7(2,1)) = 1- pq/(l + p) = q + 2p2/(1 + p). Thus Ap(7C) > Ap( 7(2,1)) if (p+ 1)(p(d+2) -1) > 2p21GI; in order for the graph G to be useful there needs to be a value of p for such that this inequality holds, and this requires (d + 3)2 > 81GI. Finding a graph G with these properties is not too easy; the constant 8 is just too large for random constructions or for standard pseudo-random constructions with d around IGI 1/ 2. But the beautiful construction of bipartite Ka,a-free graphs by Brown [14] does work. For each odd prime power r, the construction gives a graph G with IGI = 2r a and which is (r 2 - r)-regular. The required inequality (d + 3)2 > 81GI holds for r ~ 19. When r = 19, the inequality Ap (7C) > Ap ( 7(2,1)) holds in the range 1/219::; p::; 1/124. For each r ~ 23, it can be checked that the inequality holds at p = 1/12r and at p = 1/6r, and thus for all p between these values. By Bertrand's postulate this means that, for every p ::; 1/124, there is some r for which Brown's graph G satisfies Kp(H) ~ Ap(7C) > Ap( 7(2,1)) . In the light of the preceding discussion, it would be interesting to know for which values of p it is true that Kp(H) = Ap( 7(2 ,1)). We describe an approach for reducing the bound p ~ 1/3 of Example 5.18, which could be applied to other extremal problems as well. We suppose from now on that p ::; 1/3 and that 7(2,1) is not optimal; that is, there is another p-core type 7 which is optimal, and Ap ( 7) > Ap ( 7(2,1)) = 1 - pq/(l + p). By Lemma 5.14, all the vertices of 7 are red, and its edges are green and blue. Consider a sequence of graphs (Gn) of maximal weight in Q(7), as described in Remark 3.6. We have liminf bp(Gn)/n = Ap(7) > 1-pq/(1+p). Notice that the largest blue complete subgraph in any graph G" has at most 171 vertices, because all the vertices of 7 are red. In particular, the type 7(0,1) , consisting of a single blue vertex, is not contained in (Gn), and so any type that is contained in (Gn ) can have only red vertices. Moreover,

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281

by Remark 5.15, no type contained in (Gn ) contains a green triangle or a green K 3 ,3 . We define two special types, F(t) and G(k). All the vertices of both types are red . The type F(t) has t + 1 vertices, and contains a green star KI ,t, all other edges being blue . Thus F(t) is the graph HI of Example 5.12 with b = t and all vertices coloured red. The graph G(k) has k + 2 vertices, and contains a green K 2,k , all other edges being blue . For convenience we label the vertices of F(t) with 1, . .. , t + 1 so that the green edges meet vertex 1. Similarly, we label G(k) with 1, ... , k + 2 so the green edges meet vertices 1 and 2.

If (Gn) contains F(t) or G(k) we apply Lemma 3.11 to obtain an extension a, By choosing the vector x carefully, we can arrange that the new vertex in the extension is joined to the rest of the type in a way that is useful to us. We consider the case that (Gn) contains G (k) first because, though the case F(t) is easier, it makes use of the case G(k) .

Claim 1. (Gn) does not contain G(k) for k

~ l/p -

3.

Proof. If the claim fails then, for any vector x of our choosing, Lemma 3.11 gives an extension o of G(k) contained in (Gn) , with 'L.j::i Xjw(jv) > 1-pq/(1+p) . As observed above , the vertices of a must all be red . We call a vertex j E {I, .. . , k + 2} a green neighbour if vj is green . Since (Gn) contains a, a cannot contain a green triangle or a green K3 ,3 . In particular, the sets {1,2} and {3, .. . , k + 2} cannot both contain a green neighbour. We choose a number 0 :S a :S 11k and put Xj = a for j ;::: 3, X l = X2 = (1 - ka)/2 , so x E ~; the exact choice of a remains to be decided. First note that we must have a green neighbour, otherwise 'S~::i Xjw(jv) 1 - pq/(l + p) . Suppose the event occurs that the nui iber of green neighbours in {3, . . . , k + 2} is eit her one or two. Then 'L.~=-~ Xj w (jv ) ~ q(l- ka) + 2a + (k - 2)qa. So we can forbid this event by choosing a with q(l-ka) +2a+ (k-2)qa ~ 1-pq/(1 +p); that is, a ~ p/(l+p). The event that {3, .. . ,k + 2} contains at least three green neighbours means that at least one of vI or v2 is red, else a would contain a green K 3 ,3 , so we can forbid this event too by choosing (p+q)(l- ka)/2 + ko ~ 1- pq/(l + p) , or kcc ~ 1- 2pq/(1 + p). Thus choosing a to satisfy both these upper bounds ensures that {3, .. . , k + 2} contains no green neighbours. ~ q ~

We can forbid the event that exactly one of the vertices in {1,2} is a green neighbour by choosing a with (1+q)(1-ka)/2+kqa ~ 1-pq/(1+p) , or ko: ~ (1 - 3p)/(1 + p). If we can further choose a so that (1 - ka) +

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(k - l)qa + pa ::; 1 - pql(l + p), or (1 + (k - 2)p)a ~ pql(l + p), then we can also forbid the event that both 1 and 2 are green neighbours and some vj, j E {3, .. . , k + 2}, is red. Therefore a choice of a that satisfies both of these lower bounds as well will guarantee that a = G(k + 1). Finally, if we can choose an a that also satisfies the additional bound (1 - ka) + kqo. ::; 1 - pql(l + p), or ka ~ ql(l + p), then we obtain a contradiction; the inequality l:;~i Xjw(jv) > 1 - pql(l + p) becomes impossible.

If lip - 3 ::; k ::; lip - 1 we choose a = pl(l + p) ::; 1/(k + 2). Then ko: ::; (1 - p)/(l + p) ::; 1 - 2pql(1 + p), so the upper bounds for a are all satisfied. The first lower bound holds because kp ~ 1 - 3p, and so does the second because 1 + (k - 2)p ~ q. So if lip - 3 ::; k ::; lip - 1 and (Gn) contains G(k), then it also contains G(k + 1). If k ~ lip - 1 we choose a = qlk(l + p) < 11k. Now qlk ::; p and ko: = (1 - p)/(l + p) so once again the upper bounds hold . The first lower bound holds because q ~ (1 - 3p), as does the second because (1 + (k - 2)p) ~ kp. Moreover, since ko: = ql(l + p) the additional bound is satisfied, and so this time we obtain a contradiction. Therefore, if (Gn) contains G(k) and k ~ lip - 3 then, by noting that G(k) contains G(k - 1) and reducing k if necessary, we may presume that k ::; lip - 2. Then (Gn) contains G(k + 1) and, since k + 1 ::; lip - 1, (Gn) also contains G(k + 2). But k + 2 ~ lip - 1 and we now obtain a contradiction, so proving the claim. _ We now look further at what happens if (Gn) contains F(t) . We proceed in a similar manner to the discussion for G(k) , though the argument is much simpler.

Claim 2. If (Gn) contains F(t) then 6 + l2(t - l)plqJ

< lip.

Proof. If (Gn) contains F(t) then it contains an extension a of F(t) with l:~~~ Xjw(jv) > 1-pql(1+p) . Choosing 0::; a::; lit , we set Xl = (l-ta) and Xj = a for j ~ 2. We prevent vI being red by choosing a so that p(l - ta) + to ::; 1 - pql(l + q), or ta ::; 1/(1 + p). If in fact vI is green, then none of v j, j ~ 2 is green since a contains no green triangle, and we can guarantee that these edges are all blue by choosing a so that (1- ta) + q(t -l)a + pa ::; 1- pql(l + q), or (p(t - 1) + q) a ~ pql(l + p). This means that if v 1 is green then a contains F (t + 1). On the other hand, if ul is blue, we can ensure that at least k of the edges vj, j 2: 2 are green

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Extremal Graphs and Multigraphs with Two Weighted Colours

by choosing 0; so that q(l - to;) + (k - 1)0; + (t - k + l)qo; ::; 1- pq/(l + q), or (k - 1)0; ::; 2p/(1 + p). This last condition means that if vI is blue then (J contains G (k). We choose 0; = pq/(1+p)(p(t-1)+q) , so 0; < p/(p(t-1)+q) < l/t and to; < tp/(1+p)(p(t-1)+q) < l/(l+p), and therefore the first upper bound is satisfied. The lower bound holds by definition of 0;. Finally, the last upper bound holds provided (k - l)q/(p(t -1) + q) ::; 2, or k ::; 3 + 2(t -l)p/q. Suppose now that (Gn) contains F(t) but does not contain G(k), where k = 3 + l2(t - l)p/qJ. Then (Gn) contains F(t + 1). But then it must likewise contain F(t+2), and F(t+3), and in fact F(n) for all n. But then (Gn) contains arbitrarily large blue cliques, contradicting the fact noted above that the largest blue clique can have at most ITI vertices. Hence if (Gn) contains F(t) for some t, then it contains G(k) where k = 3+ l2(t-1)p/qJ. But, by Claim 1, (Gn) does not contain G(k) if k ~ 1/p-3. We conclude that if (Gn) contains F(t) then 6 + l2(t - l)p/qJ < l/p. • Recall now Example 5.12. Taking b = 10 we have K p (1£ ) = max {q, 1-q/2,q+p/2+(p-q)/18}. Certainly Ap(T) > 1-pq/(1+p) > 1-p=q and Ap(T) > 1 - q/2. Moreover Ap(T) - [q + p/2 + (p - q)/18] > 1 pq/(l + p) - [q + p/2 + (p - q)/18] = (5p - 1)2/18(1 + p) ~ O. Therefore Ap(T) > K p(1£ ). Now (Gn) is a sequence of graphs in Q(T) of maximum weight, and so wp(Gn) = (Ap(T) + 0(1)) ( I~I) by Remark 3.6. It follows that the graphs (Gn) contain a member of 1£ if n is large. Now T has only red vertices and does not contain a green triangle. So, since en is in Q(T), it cannot contain H2, which is a green triangle. Therefore G" must contain HI. The fact that G" E Q(T) now means that T contains F(10).

So we have a contradiction to Claim 2 if 6 + l18p/qJ ~ l/p. This condition holds when p = 1/8, and so it holds for all larger values of p. The contradiction means that T(2,1) is extremal for H; thus we have reduced the bound of Example 5.18 to p ~ 1/8. We can extend this result to the interval 1/9 ::; p < 1/8 with a further small observation. It is enough to show that (Gn) contains either G(6) or F(13), because 6 + l24p/qJ ~ 9 ~ l/p. Assume , to the contrary, that T contains neither G(6) nor F(13). Now T has only red vertices and no T. The vertex is joined to more than 12 others by green , since F(13) simple argument of Lemma 5.4 shows that Ap(T) ::; q + (13p - q)/ITI. Thus 1 - pq/(l + p) < q + (13p - q)/ITI, or ITI < (1 + p)(14p - 1)/2p2 ::; (1 + 1/8)(14/8 - 1)92/2 < 35.

et

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Let x E ~ be such that xtWp(r)x = Ap(r). Given a vertex u E V(r), let N(u) = { v E V(r) : uv is green} and let d(u) = L: {Xj : j E N(u)}. By Fact 3.19, Ap(r) = (p-q)xu+q+pd(u), so d(u) = (Ap(r)-q) /p+(q/p-1)x u. Now Ap(r) - q > 1- pq/(l + p) - q = 2p2/(1 + p), and so (Ap(r) - q) [p > 2p/(1 + p) ~ 1/5 . Thus d(u) > 1/5 + (q/p - l)x u ~ 1/5 + 6xu holds for all u E V(r). Choose u with Xu maximal. In particular, Xu ~ l/lrl > 1/35. Now N (u) has at most 12 vertices so we can choose v, wEN(u) with Xv + X w ~ 2d(u)/12 = d(u)/6. Since r contains no green triangle we have N(u) n N(v) = N(u) n N(w) = 0. So, writing d = L: {Xj : j E N(v) n N(w)} we have d ~ d(v) + d(w) - (1- d(u)) ~ -2/5 + 6xu + 6(x v + x w ) ~ -2/5 + 6x u + d(u) ~ -1/5 + 12xu. But N(v) n N(w) contains at most 5 vertices because r contains no rC(6)' Therefore d :::; 5xu, giving 12xu - 1/5 :::; 5xu or Xu :::; 1/35, contradicting Xu > 1/35. In conclusion, we have shown that /'\,p( r(H)) = Ap(r(2, 1)) for p ~ 1/9. Acknowledgement. The authors are grateful to Ryan Martin for stimulating discussions and to Miklos Simonovits for pointing them towards the literature on the extremal theory of directed graphs and multigraphs.

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V. E. Alekseev, Hereditary classes and coding of graphs, Probl. Cybern., 39 (1982) 151-164 (in Russian).

[2]

N. Alan, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica, 20 (2000), 451-476 .

[3]

N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, Proc. 46th IEEE Symp . Foundations of Computer Science (2005), 429-438 .

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N. Alon and U. Stav, What is the furthest graph from a hereditary property? Random Stru ctures and Algorithms, 33 (2008),87-104.

[5]

N. Alon and U. Stav, The maximum edit distance from hereditary graph properties, J. Combinatorial Theory (Ser. B), 98 (2008), 672-697 .

[6]

N. Alon and U. Stav, Stability type results for hereditary properties, (preprint).

[7]

M. Axenovich, A. Kezdy and R. Martin, On the editing distance of graphs, J. Graph Theory, 58 (2008), 123-138.

[8]

J. Balogh and R. Martin, Edit distance and its computation, Electronic Journal of Combinatorics, 15 (2008), R20.

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J. Beck, Combinatorial games.Tic-tac-toe theory. Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press , Cambridge (2008). xiv+732pp.

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B. Bollobas, Extremal Graph Theory, Academic Press, London (1978) xx+488pp.

[11] B. Bollobas and A. Thomason, Hereditary and monotone properties of graphs, in "The Mathematics of Paul Erdos II" (R. L. Graham and J. Nesetril , eds.) Algorithms and Combinatorics, 14 Springer-Verlag (1997), 70-78 . [12]

B. Bollobas and A. Thomason, The structure of hereditary properties and colourings of random graphs, Combinatorica, 20 (2000), 173-202.

[13]

J. A. Bondy and Zs. Tuza, A weighted generalization of Turan's theorem, J. Graph Theory, 25 (1997), 267-275.

[14] W . G. Brown , On graphs that do not contain a Thomsen graph, Canad. Math. Bull., 9 (1966), 281-285. [15] W . G. Brown, P. Erdos and M. Simonovits, Extremal problems for directed graphs, J. Combinatorial Theory (Ser . B) , 15 (1973), 77-93 . [16]

W. G. Brown, P. Erdos and M. Simonovits, Inverse extremal digraph problems, "Finite and Infinite Sets, Eger (Hungary) 1981", Colloq. Math. Soc. Janos Bolyai, 37, Akad. Kiad6, Budapest (1985), 119-156 .

[17]

W. G. Brown, P. Erdos and M. Simonovits, Algorithmic solution of extremal digraph problems, Trans. Amer. Math. Soc., 292 (1985), 421-449.

[18]

W . G. Brown and M. Simonovits, Extremal multigraph and digraph problems, "Paul Erdos and his mathematics, II (Budapest, 1999)" Bolyai Soc. Math. Stud., 11 , Janos Bolyai Math. Soc., Budapest (2002), 157-203.

[19]

D. De Caen and Z. Fiiredi, The maximum size of 3-uniform hypergraphs not containing a Fano plane, J. Combinatorial Theory [Ser. B), 78 (2000),274-276.

[20]

A. Diwan and D. Mubayi, Turan's theorem with colors, (preprint) .

[21]

P. Erdos, Some recent results on extremal problems in graph theory (Results), Theory of Graphs (Internat. Sympos ., Rome , 1966), Gordon and Breach, New York; Dunod, Paris (1967), 117-123.

[22]

P. Erdos , On some new inequalities concerning extremal prope "ties of graphs. Theory of Graphs (Proc. Colloq. Tihany, 1966), Academic Press, N ~w York (1968), 77-81.

[23]

P. Erdos and M. Simonovits, A limit theorem in graph theory, Siudia Sci . Math. Hungar., 1 (1966), 51-57.

[24]

P. Erdos and A.H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc., 52 (1946), 1087-1091.

[25]

Z. Fiiredi and A. Kiindgen, Turan problems for integer-weighted graphs, J. Graph Theory, 40 (2002), 195-225.

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Z. Fiiredi and M. Simonovits, Triple systems not cont aining a Fano configuration, Combinatorics, Probability and Computing, 14 (2005), 467-484.

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Gy. Katona, T . Nemetz and M. Simonovits, On a problem of Turan in the theory of graphs (in Hungarian) , Mat. Lapok, 15 (1964) 228-238.

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[28J

P. Keevash and B. Sudakov , The Turan numb er of the Fano plane, Combinatorica, 25 (2006) , 561-574.

[29J

J. Komlos and M. Simonovits , Szemeredi 's regularity lemma and its applications in graph theory, Combinatorics, Paul Erdos is eighty, Vol. 2 (D. Miklos, V. T-Sos and T . Szonyi, eds.), Janos Bolyai Math. Soc., Budapest (1996), 295-352.

[30]

L. Lovasz and B. Szegedy, Testing properties of graphs and functions (in press) .

[31]

E. Marchant, (in preparation) .

[32J

E. Marchant and A. Thomason, The structure of hereditary properties and 2coloured multigraphs, (submitted) .

[33] H.-J . Promel and A. Steger, Excluding induced subgraphs III : a general asymptotic, Rand. Stru ct. Alg., 3 (1992) , 19-31. [34]

D. C. Richer, Ph. D. thes is, University of Cambridge (2000) .

[35J

V. Rodl and A. Sidorenko, On the jumping constant conjecture for multigraphs, J. Combinatorial Theory (Ser. A), 69 (1995), 347-357.

[36]

A. F . Sidorenko , Boundedness of optimal matrices in extremal multigraph and digraph problems , Combinatori ca, 13 (1993) , 109-120.

[37]

M. Simonovits, A method for solving ext remal problems in graph theory, stability problems, Theory of Graphs (Proc. Colloq. Tihany, 1966) , Academic Press , New York (1968) , 279-319.

[38J

M. Simonovits, Extremal graph theory, Selected topics in graph theory 2 (L. W. Beineke and R. J . Wilson , eds.) Academic Pr ess, London (1983), 161-200.

Edward Marchant

Andrew Thomason

DPMMS Centre for Mathematical Scien ces Wilberforce Road Cambridge CBS OWB

DPMMS Centre for Math ematical Sciences Wilberforce Road Cambridg e CBS OWB

UK

UK

e-mail : E. Marchant
A.G.Thomason
e-mail :

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp . 287-325.

REGULARITY LEMMAS FOR GRAPHS

VOJTECH RODL * and MATHIAS SCHACHT

Szemeredi's regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some further applications variants of the regularity lemma were considered. Here we discuss several of those variants and their relation to each other.

1.

INTRODUCTION

Szemeredi's regularity lemma is one of the most important tools in extremal graph theory. It has many applications not only in graph theory, but also in combinatorial number theory, discrete geometry, and theoretical computer science. The first form of the lemma was invented by Szemeredi [47] as a tool for the resolution of a famous conjecture of Erdos and Turan [9] stating that any sequence of integers with positive upper density must contain arithmetic progressions of any finite length. The regularity lemma roughly states that every graph may be approximated by a union of induced random-like (quasi-random) bipartite subgraphs. Since the quasi-randomness brings important additional information, the regularity lemma proved to be a useful tool. The regularity lemma allows one to import probabilistic intuition to deterministic problems. Moreover, there are many applications where the original problem did not suggest a probabilistic approach. Motivated especially by questions from computer science, several other variants of Szemeredi's regularity lemma were considered . In Section 2 'The first author was supported by NSF grant DMS 0800070.

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we focus mainly on the lemmas proved by Frieze and Karman [12] and by Alon, Fischer , Krivelevich, and M. Szegedy [21. We show how these lemmas compare to Szemeredi's original lemma and how they relate to some other variants. Most proofs stated here appeared earlier in the literature and here we just give an overview. A thorough discussion of the connections of those regularity lemmas, from an analytical and geometrical perspective was given recently by Lovasz and B. Szegedy in [30] . In Section 3 we discuss the socalled counting lemmas and the removal lemma and its generalizations. We close with a brief discussion of the limit approach of Lovasz and B. Szegedy and its relation to the regularity lemmas from Section 2. There are several surveys devoted to Szemeredi regularity lemma and its applications. The reader is recommended to consult Komlos and Simonovits [26] and Komlos, Shoukoufandeh, Simonovits, and Szemeredi [25] , where many applications of the regularity lemma are discussed. Another line of research , which we will not discuss here, concerns sparse versions of the regularity lemma. Since Szemeredi's lemma is mainly suited for addressing problems involving "dense" graphs, that is graphs with at least n( 1V1 2 ) edges, it is natural to ask for similar statements that would apply to "sparse graphs", i.e., graphs with o( 1V1 2 ) edges. It turns out that a regularity lemma applicable to certain classes of sparse graphs can be proved [22,34] (see also [1]). Such a lemma was first applied by Kohayakawa and his collaborators to address extremal and Ramsey-type problems for subgraphs of random graphs (see, e.g., [19, 20, 21]). Here we will not further discuss this line of research and we refer the interested reader to the surveys [15, 23, 31] and the references therein.

2. REGULARITY LEMMAS In this section we discuss several regularity lemmas for graphs. We start our discussion with the regularity lemma of Frieze and Karman [12] in the next section. In Section 2.2 we show how Szemeredi's regularity lemma [48] can be deduced from the weaker lemma of Frieze and Kannan by iterated applications. In Section 2.3 we discuss the (c, r) -regularity lemma, whose analog for 3-uniform hypergraphs was introduced by Frankl and Rodl [11] . We continue in Section 2.4 with the regularity lemma of Alon, Fischer , Krivelevich, and M. Szegedy [2] , which can be viewed as an iterated version of Szemeredi's regularity lemma. In Section 2.5 we introduce the regular

Regularity Lemmas for Graphs

289

approximation lemma whose hypergraph variant was developed in [37]. Finally, in Section 2.6 we briefly discuss the original regularity lemma of Szemeredi [47] for bipartite graphs and a multipartite version of it from [8] . 2.1. The regularity lemma of Frieze and Kannan The following variant of Szemeredi's regularity lemma was introduced by Frieze and Kannan [12] for the design of an efficientapproximation algorithm for the MAX-CUT problem in dense graphs. Theorem 1. For every c > 0 and every to EN there exist T FK = TFK(c, to) and no such that for every graph G = (V, E) with at least IVI = n 2: no vertices the following holds. There exists a partition VI U...Uvt = V such that

(i) to :S t :S TFK, (ii)

IVII :S ... :S Ivtl :S IVII + 1, and

(iii) for every U

~

V

where e(U) denotes the number ofedges contained in U and d(Vi, tj) = e(Vi, tj)/( IVilltjl) denotes the density of the bipartite graph induced on Vi and Vj.

Definition 2. A partition satisfying property (ii) of Theorem 1 will be called equitable and a partition satisfying all three properties (i)-(iii) will be referred to as (s, to,TFK)-FK-pariition. Sometimes we may omit to and TFK and simply refer to such a partition as c-FK-pariition. The essential properties of the partition provided by Theorem 1 are properties (i) and (iii). Property (i) bounds the number of partition classes by a constant independent of G and n and, roughly speaking, property (iii) asserts that the number of edges of any large set U can be fairly well approximated by the densities d(Vi, Vj) given by the partition VI U...Uvt = V . More precisely, e(U) ~ e(U') for any choice of U and U' satisfying for example IU n Vii ~ IU' n Vii for all i E [t]. Moreover, we note that conclusion (iii) can be replaced by the following:

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V. Rodl and M. Schacht

(iii') for all (not necessarily disjoint) sets U,W (2)

le(U, W) -

t

L

~

V

d(V; ,1'j)IU n V;IIW n I'jll ,,6m',

i=l jE[tJ\{i}

where edges contained in

un Ware counted twice in e(U, W).

Indeed, if (iii) holds, then we infer (iii') from the identity

e(U, W) = e(U U W) - e(U) - e(W)

+ 3e(U n W) .

The proof of Theorem 1 presented here relies on the index of a partition, a concept which was first introduced and used by Szemeredi. Definition 3. For a partition P = (VI, . . . , vt) of the vertex sets of a graph G = (V, E), i.e., VI U.. . uvt = V we define the index of P by t-l

ind (P)

t

~ (I~I) ~j~' d'(V;, I'j) !V;II I'j I·

Note that it follows directly from the definition of the index that for any partition P we have o::; ind (P) ::; 1. For the proof of Theorem 1 we will use the following consequence of the Cauchy-Schwarz inequality. Lemma 4. Let 1 ::; M < N , let 0"1, .. . , O"N be positive and ds , . . . , dN , and d be reals. If L:f:l O"i = 1 and d = L:f:l diO"i then

For completeness we include the short proof of Lemma 4. Proof. For M

(3)

= 1 and N = 2 the statement follows from the identity

291

Regularity Lemmas for Graphs

For general 1 :::; M

< N we infer from the Cauchy-Schwarz inequality

applied twice in the form

(I: dWi) 2:::; I: d;(Ji I: a;

N

M

N

i=1

i=1

i=M+I

L d;(Ji = L d;(Ji + L

d;(Ji

Setting M

1T1 =

L (Ji, i=1

A

we have d =

N I:i=1 dWi =

N

1T2 =

L

a.,

i=M+1

d and from (3) we infer

which is what we claimed. _ After those preparations we prove Theorem 1. Proof of Theorem 1. The proof is based on the following idea already present in the original work of Szemeredi. Starting with an arbitrary equitable vertex partition Po with to classes, we consider a sequence of partitions Po ,PI, . .. such that Pj always satisfies properties (i) and (ii). As soon as Pj also satisfies (iii) we can stop. On the other hand, if Pj does not satisfy (iii) we will show that there exists a partition Pj+1 whose index increased by £2/2. Since ind (P) :::; 1 for any partition P , we infer that after at most 2/£2 steps this procedure must end with a partition satisfying properties (i), (ii), and (iii) of the theorem.

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V. Rod] and M. Schacht

So suppose Pj = P = (VI, .. . , Vi) is a partition of V which satisfies (i) and (ii), but there exists a set U ~ V such that (1) fails. We are going to construct a partition R = Pj+I satisfying

(4)

ind (R) 2:: ind (P) + e2/2.

For that set

ti, = Vi n U and o; = Vi \ U. We define a new partition Q by replacing every vertex class Vi by U, and Ui

Next we show that the index of Q increased bye 2 compared to ind (P). For every 1 :::; i < j :::; t we set

Since we may assume t 2:: to 2:: l/e, which yields I:~=I e(Vi) :::; en 2 /2, we infer from the assumption that (1) fails, that

Since Vi = Ui.JUi for every i

E

[t] we obtain

d(Vi, l-j)IViIIl-j1 = au; Uj)IUiIIUjl + d(Ui, Uj)IUiIIUjl

+ d(Ui, Uj)IUiIIUjl + d(Ui, Uj)IUiIIUjl and Combining those identities with Lemma 4, we obtain

293

Regularity Lemmas for Graphs

Summing over all 1 :::; i (6)

ind ( Q) 2: ind (P)

<j

:::; t we obtain

+ (~) ~ Crj IUi II u, I 2

~<J

>. d (P) + (I:i<j cijlUillUjl)2 ~ . d (P) + {cn > . d (P) + 2 - III (~) I:i<j lUill Ujl - III (~) (~) - III C . 2/2)2

We now find an equitable partition R which is a refinement of P (and almost a refinement of Q) for which (4) holds. For that subdivide each vertex class Vi of P into sets Wi,a of size c2n/{5t)J or c2n/{5t)J + 1 in such a way that for all but at most one of these sets either Wi,a ~ U, or Wi,a ~ Ui holds. For every i E [t] let Wi,O denote the exceptional set if it exists and let Wi,O be arbitrary otherwise. Let R be the resulting partition. Moreover, we consider the partition R* which is a refinement of R obtained by replacing Wi,O by possibly two classes o, n Wi,O and Ui n Wi,o. Since the contribution of the index of Rand R* may differ only on pairs with at least one vertex in Wi,O for some i E [t] and since IWi,ol :::; c2n/{5t)J + 1 for every i E [t] we infer that

l

l

l

ind (R*) - ind (R):::;

tt (2

(2)-1 n

t

C

)

n

2

C

5t + 1 n:::; 2 '

for sufficiently large ti . Furthermore, since R * is a refinement of Q it follows from the Cauchy-Schwarz inequality that ind (Q) ~ ind (R*) and, consequently, ind (R) 2: ind (R*) -

C2

c2

(6)

2 2: ind (Q) - 2 2:

ind (P)

c2

+2'

which concludes the proof of the theorem. _ The proof of Theorem 1 shows that choosing

suffices. In fact, in each refinement step we split the vertex classes Vi into at most l5/c2 + lJ :::; 6/c 2 classes Wi,a, when we construct R. Hence, each time property (iii) fails the number of vertex classes of the new partition increases by a factor of 6/e 2 and in total there are at most 2/c2 iterations.

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On the other hand, it was shown by Lovasz and B. Szegedy [30] that for every < e ::; 1/3 there are graphs for which every partition into t classes satisfying property (iii) of Theorem 1 requires t 2:: 21/ (8f:) /4 and, hence, t »l/e. As a consequence Theorem 1 does not allow to obtain useful bounds for e(UnVi,UnVj), since for such a graph en 2 » n 2 /t2 = IViIIVj!. Property (iii) of Theorem 1 only implies e(U n Vi,U n Vj) ~ d(Vi ,Vj)IU n ViII U n Vj I on average over all pairs i < j for every "large" set U. However, Szemeredi's regularity lemma (which was proved long before Theorem 1) allows to control e(U n Vi, un Vj) for most i < j . The price of this is, however, a significantly larger upper bound for the number of partition classes t.

°

2.2. Szemeredi's regularity lemma In this section we show how Szemeredi's regularity lemma from [48] can be obtained from Theorem 1 by iterated applications. For that we consider the following simple corollary of Theorem 1, which was first considered by Tao [49].

Corollary 5. For all t/, e > 0, every function 8 : N -+ (0,1], and every to E N there exist To = To( u,e, 8(') , to) and no such that for every graph G = (V, E) with at least IVI = n 2:: no vertices the following holds. There exists a vertex partition P = (Vi)iE[t] with VIU ... Uvt = V and a refinement Q = (Wi,j )iE[tj, jE[s] with Wi,l U. .. UWi,s = Vi for every i E [t] such that

(i) P is an (e, to,To)-FK-partition,

(ii) Q is a (8(t), to,To) -FK-partition, and (iii) ind (Q) ::; ind (P) + v , Before we deduce Corollary 5 from Theorem 1, we discuss property (iii). Roughly speaking, if two refining partitions P and Q satisfy property (iii), then this implies that d(Wi,a, Wj,b) and d(Vi, Vj) are "relatively close" for "most" choices of i < j and a, b E [8]. More precisely, we have the following, which was already observed by Alon, Fischer, Krivelevich , and M. Szegedy [2].

Lemma 6. Let 'Y, v > 0, let G = (V, E) be a graph with n vertices, and for some positive integers t and 8 let P = (Vi)iE[tj with VIU... Uvt = V

295

Regularity Lemmas for Graphs

be a vertex partition and let Q = (Wi,j )iE[tj, jE[S] be a refinement with Wi,lU . .. UWi,s = Vi for every i E [t] . If ind (Q) :s; ind (P) + u, then

L L

1~i<j9 a,bE[s]

{IWi ,aIIWj,bl : I d(Wi,a, Wj,b) - d(Vi, Yj)1

,

Proof. For 1 :s; i < j d(Vi, Yj) ~ ,}. Since d(Vi, Yj)IViIIYjI =

~,}:s; v2 n 2 .

:s; t let

L

At

= {(a, b)

[8] X [8] : d(Wi,a, Wj,b) -

E

d(Wi,a, Wj,b)IWi,aIIWj,bl

a,bE[s]

L

L

d(Wi,a,Wj,b)IWi,aIIWj,bl +

d(Wi,a, Wj,b)IWi,aIIWj,bl,

(a,b)~At

(a,b)EAt

we obtain from the defect form of Cauchy-Schwarz (Lemma 4), that

L

i,a, Wj,b)IWi,aIIWj,b! ~ d

2(W

d

2(Vi,

Yj)IViIIYjI+,2

a,bE[s]

L

IWi,aIIWj,bl·

(a,b)EAt

Summing over all 1 :s; i

< j :s; t we get 2

ind (Q) ~ ind (P) + (:) 2

Since, by assumption ind (Q)

L

l:

IWi,aIIWj,bl ·

l Si<j St (a,b)EAt

:s; ind (P) + t/, we have

Repeating the argument with the appropriate definition of claim. _

Ai:; yields the

Proof of Corollary 5. For the proof of the corollary we simply iterate Theorem 1. Without loss of generality we may assume that o(t) :s; c for every tEN. For given u, c, 0(') ' and to, we apply Theorem 1 and obtain an (s, to , To)-FK-partition P with t classes. Since in the proof of Theorem 1 the initial partition was an arbitrary equitable partition, we infer that after

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V. Rod] and M. Schacht

another application of Theorem 1 with t5(t) (in place of c) and to we obtain an equitable refinement Q of P which is a (t5(t), to, To)-FK-partition with st classes. In other words, P and Q satisfy properties (i) and (ii) of Corollary 5 and if (iii) also holds, then we are done. On the other hand, if (iii) fails, then we replace P by Q and iterate, i.e., we apply Theorem 1 with t5(ts) (in place of c) and to = ts to obtain an equitable refinement Q' of P' = Q. Since we only iterate as long as (iii) of Corollary 5 fails and since 1/ is fixed throughout the proof, this procedure must end after at most 1/1/ iterations. Therefore the upper bound To on the number of classes is in fact independent of G and n and can be given by a recursive formula depending on 1/, e, 15(·), and to. • We now show that Corollary 5 applied with the right choice of parameters yields the following theorem, which is essentially Szemeredi 's regularity lemma from [48].

Theorem 7. For every e > 0 and every to E N there exist Ts z = Ts z (s, to) and no such that for every graph G = (V, E) with at least IVI = n 2: no vertices the following holds. There exists a partition VI U...uvt = V such that

(i) to ::; t ::; Ts z , (ii)

IVII::; ..·::; Ivtl ::; IVII + 1, and

(iii) for all but at most ct2 pairs (Vi , ltJ) with i < j we have that for all subsets ii, ~ Vi and u, ~ ltJ

(7) We note that the usual statement of Szemeredi's regularity lemma is slightly different from the one above. Usually clVillltJl on the right-hand side of (7) is replaced by clUillUjl and for (iii) it is assumed that lUiI 2: ciVil and IUjl 2: clltJl. However, applying Theorem 7 with c' = c3 would yield a partition with comparable regular properties.

Definition 8. Pairs (Vi, ltJ) for which (7) holds for every o, ~ Vi and l'J are called e -reqular. Partitions satisfying all three properties (i)(iii) of Theorem 7, we will refer to as (c, to, Tsz)-Szemenidi-partition. Again we may sometimes omit to and Ts z and simply refer to such partitions as e-Szemeredi-pcrtiiums. Uj ~

297

Regularity Lemmas for Graphs

Below we deduce Theorem 7 from Corollary 5 and Lemma 6.

Proof of Theorem 7. For given e > 0 and to, we apply Corollary 5 with and

t~ = to

and obtain constants T~ and n~ which define Tsz = T~ and no = n~ . (We remark that the choice for e' has no bearing for the proof and therefore we set it equal to 1.) For a given graph G = (V, E) with n vertices Corollary 5 yields two partitions P = (Vi) iE[t] and Q = (Wi,j )iE[tj, jE[S] satisfying properties (i)-(iii) of Corollary 5. We will show that, in fact, the coarser partition P also satisfies properties (i)-(iii) of Theorem 7. Since P is an (c:', t~ , T~)-FK-partition by our choice of t~ = to and Tsz = T~ the partition P obviously satisfies properties (i) and (ii) of Theorem 7 and we only have to verify property (iii) . For that we consider for every 1 ::; i < j ::; t the set

and we let 1= {

{i,j} : 1::; i < j ::; t such that

I:

IWi,aIIWj,bl

2::

£IViIIVjI/6}.

(a,b)EAij

We will first show that III :::; ct 2 and then we will verify that if {i, j} t/: I, then (7) holds. Indeed, due to property (iii) of Corollary 5 we have ind (Q) ::; ind (P) + 1/' and , consequently, it follows from Len rna 6 (applied with 1/' = £4/362 and "(' = c:/6) that

2::

I: I: {i,j}EI (a,b)EAij

IWi,aIIWj,bl

2::

~

I:

IViIIVjI·

{i ,j}EI

Moreover, since IViI 2:: In/tJ 2:: n/(2t) for every i E [t] we have c:n 2/6 2:: IIln 2 / (4t2 ) and, consequently,

(8)

298

V. Rod] and M. Schacht

Next we will show that if {i,j} tI. I then the pair (Vi, Vj) is s-regular, i.e., we show that (7) holds for every ti, ~ Vi and ~ Vj. For given sets u, ~ Vi and ~ Vj and a, b E [s] we set

o,

o,

U'b J , = UJ· n WJ,'b

and and have

L

e(Ui, Uj) =

e(Ui,a, Uj,b)'

a,bE[s]

Appealing to the fact that Q is a (<5'(t), to, To) -FK-partition we obtain from (2) that e(Ui, Uj) =

L

d(Wi,a, Wj,b)IUi,aIIUj,bl

± 6<5'(t)n 2 .

a,bE[s]

From the assumption {i, j}

L

tI. I

we infer

L

d(Wi,a, Wj,b)IUi,aIIUj,bl:S

(a,b)EA;j

(a,b)EA;j

and, furthermore, for (a, b)

tI. Aij

IWi,aIIWj,bl:S ~IViIIVjI

we have

Combining, those three estimates we infer e(Ui, Uj) =

L

d(Vi, Vj)IUi ,aIIUj,bl ±

~IUiIIUjl ± ~IViIIVjI ± 6<5'(t)n 2 •

a,bE[s]

Hence from our choice of <5' (t) and Vi 2: ln/ t J 2: n/ (2t) we deduce

which concludes the proof of Theorem 7. • In contrast to Theorem 1 the upper bound Ts z = Ts z (€, to) we obtain from the proof of Theorem 7 is not exponential, but of tower-type. In fact, we use Corollary 5 with 1/ = f4/36 2 and 8(t) = f/(36t2). Due to the choice of 1/ we iterate Theorem 1 at most 362 / f4 times and each time the number of classes grows exponentially, i.e., ti classes from the i-th iteration may split

299

Regularity Lemmas for Graphs

2 into 20 (tt /c ) classes for the next step. As a consequence, the upper bound Tsz = TSz(e, to), which we obtain from this proof, is a tower of 4's of height O(c 4 ) with to as the last exponent. The proof of Szemeredi's regularity lemma from [48] yields a similar upper bound of a tower of 2's of height proportional to C 5 . However, recall that the statement from [48] is slightly different from the version proved here, by having a smaller error term in (7). A lower bound of similar type was obtained by Gowers [17]. In fact , Gowers showed an example of a graph for which any partition satisfying even only a considerably weaker version of property (iii) requires at least t classes, where t is a tower of 2's of height proportional to l/e 1/ 16 .

2.3. The (e,r)-regularity lemma As we have just discussed in the previous section, the example of Gowers shows that we cannot prevent the situation when the number of parts t of a Szemeredi-partition is much larger than, say, l/e. For several applications this presents an obstacle which one would like to overcome. More precisely one would like to obtain some control of the densities of subgraphs which are of size much smaller than, say, n/t 2 . The (e, r)-regularity lemma (Theorem 9), the regularity lemma of Alon, Fischer , Krivelevich, and M. Szegedy (Theorem 10), and the regular approximation lemma (Theorem 11), were partly developed to address such issues. A version for 3-uniform hypergraphs of the following regularity lemma was obtained by Frankl and Rodl in [l1J.

Theorem 9. For every e > 0, every function r : N -+ N, and every to E N there exist TFR = TFR( e, r(·), to) and no such that for every graph G = (V, E) with at least IVI = n ~ no vertices the following holds. There exists a partition VIU...U"\tt = V such that

(i) to ::; t ::; TFR , (ii)

1V11::;·· ·::; l"\ttl::; IVII + 1, and

(iii) for all but at most et 2 pairs (Vi, ltj) with i < j we have that for all TT d Uj1 , ... , Ur(t) V sequences 0 f su bsets ini" '" Ur(t) i ~ v i an j ~ j

rW

(9)

Iq~ E(U;',UJ)I-

rW d(V; , V;)I!d u;' x

UJI ,,£IV;IIV;I,

300

V. Rod] and M. Schacht

Note that if r(t) == 1 then Theorem 9 is identical to Theorem 7 and if r(t) == k for some constant kEN (independent of t), then it is a direct consequence of Theorem 7. We remark that for arbitrary functions r( ·), Theorem 9 can be proved along the lines of Szemeredi's proof of Theorem 7 from [48]. Below we deduce Theorem 9, using a slightly different approach, namely we infer Theorem 9 from Corollary 5 in a similar way as we proved Theorem 7.

Proof. For given c, r(·), and to we follow the lines of the proof of Theorem 7. This time we apply Corollary 5 with a smaller choice of 8'(.)

c' = 1,

and

t~ =

to

and obtain T6 and n~, which determines TFR and no. We define the sets A ij and I identical as in the proof of Theorem 7, i.e., for 1 :::; i < j :::; t we set

Aj = {(a,b) E [s]

x

[s] : Id(Wi,a, Wj,b) - d(Vi, Vi)1 2:: c/6}

and we let 1= {{i,j}

1 :::; i < j :::; t such that

L

IWi,aIIWj,bl 2::

cIViIIViI/6} .

(a,b)EA i j

Again we obtain (8) and the rest of the proof requires some small straightforward adjustments. We set r = r(t) and we will show that if {i,j} ~ I, then (9) holds Al A Al A for every sequence Ui , . . . ,U[ ~ Vi and Uj, .. . ,UJ ~ Vi . For such given sequences we consider new sequences ul,..., UiR ~ Vi and UJ , . . . , Ul ~ Vi satisfying the disjointness property (see (10) below). For that let R = 4r -3 r and for a non-empty set 0 =1= L ~ [r]let

and for two sets L, L' with non-empty intersection we let Ui(L, L'} Ui(L) and Uj(L, L') = Uj(L'). Note that there are R = 4r - 3r such pairs of sets L, L' and we can relabel the sequences (Ui(L, L'}) LnL'1-0 and

(Uj(L, L'))

LnL'i'0

to

ul,··. ,UiR ~ Vi and

UJ, . . . , Ul ~

Vi . Note that for

301

Regularity Lemmas for Graphs

Ur

all p =1= q the sets and uiq may either be equal or disjoint. Moreover, due to this definition we obtain for all 1 ::; p < q ::; R (10) (U~t xU?) n (U?t x UT!) J J

UqE[R] in x U? = UUq «tn

= 0 and

t

J

J

t

qE[r]

Furthermore, for q E [R] and a, bE [s] we set U~ t ,a

= U9t n Wi,a

U?b J,

and

= U?J n WJ·'b

and we get for every q E [R]

e(Uiq,UJ) =

L

e(Ui~a,U1b)'

a,bE[s] Appealing to the fact that Q is a (8'(t), to, To) -FK-partition we obtain from (2) that

q e(Ui ,UJ) =

L

d(Wi,a, Wj,b) IUi~all U1bl ± 68'(t)n

2

•

a,bE[s] From the assumption {i, j} we infer

L L

(a,b)EA;j qE[R]

tf: I

and the disjointness property from (10)

d(Wi,a ,Wj,b) IUi~all U1bl::;

and, furthermore, for (a, b)

L

IWi,aIIWj,bl::;

(a,b)EA;j

~IViIIVjI

tf: Aij we have

for every q E [R] . Combining, those three estimates we infer

IqE[r] U E(U? ,UJ) 1= IUqE[R]E(Uiq,UJ)1 = (d(Vi, Vj) ±~)

= d(Vi , Vj)1

L L IUi~all U1bl ± ~IViIIVjI ± 6R8'(t)n

qE[R] a,bE[s]

U E(Uiq, UJ) I± ~IViIIVjI ± 6R8'(t)n 2 .

qE[R]

2

v. Rod] and M. Schacht

302 Hence from our choice of 8'(t), R = (4r deduce from (10)

-

3r ) , and Vi

~

lnjtJ

~

nj(2t) we

which concludes the proof of Theorem 9. •

2.4. The regularity lemma of Alon, Fischer, Krivelevich, and M. Szegedy In the last two sections we iterated the regularity lemma of Frieze and Karman and obtained Corollary 5, from which we deduced Szemeredi's regularity lemma (Theorem 7) and the (c, r) -regularity lemma (Theorem 9). From this point of view it seems natural to iterate these stronger regularity lemmas. This was indeed first carried out by Alon, Fischer , Krivelevich, and M. Szegedy [2] who iterated Szemeredi's regularity lemma for an application in the area of property testing.

Theorem 10. For every 1/, c > 0, every function J : N -+ (0,1]' and every to E N there exist TAFKS = TAFKS ( 1/, e, J('), to) and no such that for every graph G = (V, E) with at least IVI = n ~ no vertices the following holds. There exists a vertex partition P = (Vi)iE[t] with VI U. .. uvt = V and a refinement Q = (Wi,j )iE[tj, jE[s] with Wi,I U. . . UWi,s = Vi for every i E [t] such that

(i) P is an (s, to, TAFKs)-Szemeredi-partition, (ii) Q is a (8(t) ,to, TAFKS) -Szemetedi-pertition , and

(iii) ind (Q) S; ind (P) + 1/. Proof. The proof is identical to the proof of Corollary 5 with the only adjustment that we iterate Theorem 7 instead of Theorem 1. • The price for the stronger properties of the partitions P and Q, in comparison to Szemeredi's regularity lemma, is again in the bound TAFKS. In general TAFKS can be expressed as a recursive formula in 1/, e, J('), and

Regularity Lemmas for Graphs

303

to, and for example, if <5(t) is given by a polynomial in l/t, then TAFKS is an iterated tower-type function , which is sometimes referred to as a wowzertype function. Theorem 9 relates to Theorem 10 in the following way. It is a direct consequence of (9) that if (lIi, Vj) is not one of the exceptional pairs in (iii) of Theorem 9, then for any partition of lIi and Vj into at most .jifi) parts of equal size, "most" of the r(t) pairs have the density "close" (up to an error of 0 = (JE)) to d(lIi, Vj). Hence, if we set at the beginning r(t) =

(TS z ( <5(t), t))2 and then apply Theorem 7 to obtain a (<5(t), t ,Tsz ( <5(t), t))Szemeredi-partition Q, which refines the given partition, then we arrive to a similar situation as in Theorem 10. In fact, we have two Szemeredipartitions satisfying (i) and (ii) of Theorem 10 and (iii) would be replaced by the fact that d(Wi,a, Wj,b) ~ d(lIi, Vj) for "most" pairs from the finer partition Q.

2.5. The regular approximation lemma The following regularity lemma is another byproduct of the hypergraph generalization of the regularity lemma and appeared in general form in [37] . In a different context, Theorem 11 appeared in the work of Lovasz and B. Szegedy [30, Lemma 5.2] .

Theorem 11. For every v > 0, every function 6 : N -+ (0,1], and every E N there exist To = To(v, 6(-), to) and no such that for every graph G = (V, E) with at least IVI = n ~ no vertices the following holds. There exists a partition P = (lIi)iE[tj with VIU.. .Uyt = V and a graph H = (V, E') on the same vertex set V as G such that

to

(a) P is an (6(t), to,To) - Szemeredi-pertition for H and

(b)

IELE'I = IE \ E'I + IE' \ EI :::; vn 2 .

The main difference between Theorem 11 and Theorem 7 is in the choice of 6 being a function of t. As already mentioned, it follows from the work of Gowers [17] (or alternatively from the work of Lovasz and B. Szegedy [30, Proposition 7.1]) that it is not possible to obtain a Szemeredi (or even a Frieze-Kannan) partition for certain graphs G with 6 of order l/t. Property (a) of Theorem 11 asserts, however, that by adding and deleting at most vn 2 edges from/to G we can obtain another graph H which admits a "much more" regular partition, e.g., with 6(t ) « l/t.

304

V. Rod] and M. Schacht

Below we show how Theorem 11 can be deduced from the iterated regularity lemma of Alon, Fischer, Krivelevich and M. Szegedy (Theorem 10). The idea is to apply Theorem 10 with appropriate parameters to obtain Szemeredi-partitions P = (Vi)iE[t] and Q = (Wi,j )iE[t] , jE[s] for which Q refines P and ind (Q) ::::; ind (P) + v' , The last condition and Lemma 6 imply that d(Wi,a, Wj,b) ~ d(Vi, Vj) (with an error depending on v') for "most" i < j and a, b E [8]. The strong regularity of the finer partition Q will then be used to adjust G (by adding and removing a few edges randomly) to obtain H for which P will have the desired properties. We now give the details of this outline . Proof of Theorem 11. For given u, e('), and to we apply Theorem 10 with v' = v 3/16, some arbitrary e', say e' = 1, 8'(t) = min {e(t)/2,v/4} , and t~ = to. We also fix an auxiliary constant " = v /2. We then set To = TAFKS and no = n~ . After we apply Theorem 10 to the given graph G = (V, E), we obtain an (s', to, To)-Szemeredi-partition P and a (8' (t ),to, To) - Szemeredipartition Q which refines P such that ind (Q) ::::; ind (P) + v', Next we will change G and obtain the graph H, which will satisfy (a) and (b) of Theorem 11. For that:

(A) we replace every subgraph G[Wi,a , Wj,b] which is not 8'(t)-regular by a random bipartite graph of density d(Vi, Vj) and

(B) for every 1 ::; i < j ::::; t and a, b E [8] we add or remove edges randomly to change the density of G[Wi,a, Wj,b] to d(Vi, Vj) + 0(1) . It follows from the Chernoff bound that the resulting graph H = (V, E ') has the property that for every 1 ::; i < j ::::; t and a, b E [8] the induced subgraph H[Wi,a,Wj,b] is (8' (t )+ 0(1)) -regular and dH(Wi,a, Wj,b) = dH(Vi, Vj)+0(1), where 0(1) -t 0 as n -t 00. (Recall that for G from Lemma 6 we can only infer that da(Wi,a ,Wj,b) = da(Vi, Vj) ±,' for "most" pairs for some " »8' (t ).) Hence for every 1 ::::; i < j ::::; t and arbitrary sets U, ~ Vi and ti, ~ Vj we have

=

z=

a,bE[s]

(dH(Wi,a, Wj,b) lUi n Wi,allUj n Wj,bl ±

(8' (t ) + 0(1)) IWi,aIIWj,bl)

305

Regularity Lemmas for Graphs

In other words, the partition P is a (20'(t) :s; c(t),to, To) -Szemeredipartition for H, which is assertion (a) of Theorem 11. For part (b) we will estimate the symmetric difference of E and E'. Since Q is a (0'(t), to, To) Szemeredi-partition for G the changes in Step (A) contributed at most

(11) to that difference. For estimating the changes introduced in Step (B) we appeal to Lemma 6. From that we infer that, since ind (Q) :s; ind (P) changed only

+ o', we

"typically"

,'IWi,aIWj,bl pairs . More precisely, in Step (B) we changed at most (12)

L L

,'IWi,aIIWj,b l

i<j a,bE[s]

+

L L

{IWi ,aIIWj,bl

i<j a,bE[s]

v') ("2 (,,)2

Idc(Wi,a,Wj,b) - dc(Vi, Vj)1 ~ ,'}

v

< _ + __ n 2 < _n 2 .

-

- 2

Finally, from (11) and (12) we infer IE6E'1 satisfies property (b) of Theorem 11. •

:::;

vn 2 ,

which shows that H

2.6. An early version of the regularity lemma

We close this section with the statement of an early version of Szemeredi's regularity lemma, which was introduced in [47] and one of the key components in the proof of the Erdos-Turan conjecture concerning the upper density of subsets of the integers containing no arithmetic progression of fixed length. Another application of that lemma lead to the upper bound for the Ramsey-Turan problem for K4 due to Szemeredi [46] and to the resolution of the (6,3)-problem, which was raised by Brown, Erdos and Sos [6, 45J, and solved by Ruzsa and Szemeredi [43J .

306

V . ROdl and M. Schacht

Theorem 12. For all positive CI , C2, 0, e. and a there exist To, So, M, and N such that for every bipartite graph G = (XUY, E) satisfying IXI = m ~ M and IYI = n ~ N there exists a partition XOUXIU ... UXt = X with t ~ To and for every i = 1, . . . ,t there exists a partition Yi,OUYi,1U. .. UYi,si = Y with s, ~ So such that

(a) IXol ~

em and IYi,ol ~ an for every i =

1, . .. , t, and

(b) for every i = 1, ... , t, every j = 1, ... , Si, and all sets U ~ Xi and W ~ Yi,j with lUI ~ cIIXi ! and IWI ~ c2lYi,jl we have d(U,W) > d(Xi, Yi,j) - o. •

Note that this lemma does not ensure such an easy to use structure of the partition as the later lemmas. More precisely, the partitions of Y may be very different for every i = 1, . .. ,t. On the other hand , the upper bounds To and So are of similar type as those of Theorem 1, i.e., we have To, So = 2poly(l/min{cI ,c2,8,g,er}). We also point out that for example in [43] Theorem 12 was applied iteratively, which in turn lead to a tower-type bound for the (6,3)-problem. A multipartite version of Theorem 12 was developed by Duke, Lefmann, and Rodl [8] for efficiently approximating the subgraph frequencies in a given graph G on n vertices for subgraphs of up to n( Jlog log (n) ) vertices. Theorem 13. For every e > 0 and every integer k ~ 2 there exist To = 4k 2 / c5 such that for every k-partite graph G = (V, E) with vertex classes VIU UVk = V and IVII = . . . = IVkl = N there exists a partition P of VI x X Vk such that (i) the number of elements WI x

(ii) IWil ~ c k2/ c N for every i and 5

=

X

1,

Wk in P is at most To,

, k and every WI x ... x Wk in P,

(iii) we have

L

k

II IWil ~ cN

k

.

i=1 for the subfamily Pirr ~ P containing those elements WI x .. . X Wk from P which contain an irregular pair (Wi, Wj), i.e., a pair (Wi, Wj) with i < j for which there exist subsets ti, ~ Wi and u, ~ Wj with IUil ~ clWil and IUjl ~ c!Wjl such that Id(Ui , Uj) - d(Wi , Wj)1 > e. WI X" 'XWkE'Pirr

•

Regularity Lemmas for Graphs

307

The main advantage of Theorem 13, in comparison to Szemeredi's regularity lemma (Theorem 7), is the smaller upper bound To. The partition in Theorem 13 still conveys information if l/e and k tend slowly to infinity with n = lVI, for example, if l/e and k are of order log'{n) for some small constant c> O. Due to the tower-type bound of Theorem 7 there l/e can be at most of order log"(n), where log" denotes the iterated logarithm function. On the other hand, the upper bound To in Theorem 13 is comparable to the one from Theorem 1 and as we will see in the next section Theorem 1 would be also well suited for the main application of Theorem 13 in [8]. Moreover, the structure of the partition provided by Theorem 1 seems to be simpler and easier to work with.

3.

REDUCED GRAPH AND COUNTING LEMMAS

In this section we show how regular properties of the partitions given by the regularity lemmas from Section 2 can be applied to approximate the number of subgraphs of fixed isomorphism type of a given graph G. More precisely, for graphs G and F let N F (G) denote the number of labeled copies of F in G. Roughly speaking, we will show that N F (G) can be fairly well approximated by only studying the so-called reduced graph (or cluster-graph) of a regular partition. Definition 14. Let a partition of V.

E

> 0, G = (V, E) be a graph, and let P =

(Vi)iE[tj

be

(i) If P is an E-FK-partition, then the reduced graph R = Rc(P) is defined to be the weighted, complete, undirected graph with vertex set V(R) = [t] and edge weights wR(i,j) = d(Vi, lj).

(ii) IfP is an e-Szemeredi-partition, then the reduced graph R = RC(P,E) is defined to be the weighted, undirected graph with vertex set V (R) = [tl, edge set E(R) = {{i,j} : (Vi,lj) is s-regular}, and edge weights wR(i,j) = d(Vi, lj). The reduced graph carries a lot of the structural information of the given graph G. In fact, in many applications of the regularity lemma, the original problem for G one is interested in can be turned into a "simpler" problem for the reduced graph.

308

V. Rod] and M. Schacht

Remark 15. Below we will consider (labeled) copies FR of a given graph F in a reduced graph R. If R is the reduced graph of an FK-partition, then R is an edge-weighting of the complete graph and, consequently, any ordered set of V(F)I vertices of V(R) spans a copy of F. On the other hand, if R is the reduced graph of an s-Szemeredi-partition, then R is not a complete graph and for a labeled copy FR of F in R with V (FR) = {il , ... , ie} we will have that (Vi j ' Vi k ) is s-regular for every edge {ij,ik} E E(FR).

I

3.1. The global counting lemma Here by a counting lemma we mean an assertion which enables us to deduce directly from the reduced graph some useful information on the number NF(G) of labeled copies of a fixed graph F in a large graph G. We will distinguish between two different settings here. The first counting lemma will yield an estimate on NF(G) in the context of Theorem 1. Since NF(G) concerns the total number of copies, we regard this result as a global counting lemma. In contrast, for an f-vertex graph F the local counting lemma (Theorem 18) will yield estimates on NF( G[Vil' ... ' Vii]) for an induced f-partite subgraph of G given by the regular partition P. However, for such a stronger assertion we will require that P be a Szemeredi-partition,

Theorem 16. Let F be a graph with vertex set V(F) = [fl. For every there exists c > 0 such that for every G = (V, E) with IVI = nand every c-FK-partition P = (Vi)iE[tj with reduced graph R = Rc(P) we have

, >0 (13)

NF(G) =

L

II

FR {ij,ik}EE(FR)

wR(ij, ik)

II

IVi j I ± ,ni,

ijEV(FR)

where the sum runs over all labeled copies FR of F in R (cf. Remark 15).

For a simpler notation we denote here and below the vertices V(FR) of a given copy of FR of the f-vertex graph F in R by {il, ... ,ie} and omit the dependence of FR.

Proof. We follow an argument of Lovasz and B. Szegedy from [29]. We prove Theorem 18 by induction on the number of edges of F. Clearly, the theorem holds for graphs with no edges and for graphs with one edge it follows from the definition of c-FK-partition with e = ,.

309

Regularity Lemmas for Graphs

For given F and, we let e ::; ,/12 be sufficiently small, so that the statement for the induction assumption holds with " = ,/2. For two vertices x, y E V we set

dp(x,y) =

o {d(Vi, l-j)

if x, y E Vi for some i E [t], if x E Vi and y E l-j for some 1 ::; i

<j

::; t

and we denote by llE(X, y) the indicator function for E, i.e., llE(X , y) equals 1 if {x, y} E E and it equals 0 otherwise. We consider the difference of the left-hand side and the main term of the right-hand side in (13) and obtain

(14)

!NF(G) -

II

L

wR(ij, ik)

FR {ij,idEE(FR)

L

=

Xl, ...,XeE(V)e

( II {i,j}EE(F)

II IVi II j

ijEV(FR)

llE(Xi,Xj) -

II

dp(Xi,Xj)),

{i,j}EE(F)

where Xl, ... ,Xi E (V)£ is an arbitrary sequence of f. distinct vertices in V. Without loss of generality we may assume that {f. - 1, f.} is an edge in F and we denote by F- the spanning subgraph of F with the edge {f. 1,f.} removed. Then, applying the identity ala2 - f31f32 = f32(al - (31) + al(a2 - (32), we get the following upper bound for the right-hand side of the last equation (15)

2:: ( .. II

+

Xl, ...,XeE(V)e

llE(Xi,Xj)) (llE(X£-I,X£) - dp(X£_I,X£))

{t,J}EE(F-)

By the induction assumption we can bound the first term by have

,'n£, i.e., we

(16)

L Xl, ...,xeE(V)e

II

II

dp(X£-I,Xe)( . . llE(Xi,Xj) - .. dp(Xi,Xj)) {t,J}EE(F-) {t,J}EE(F-)

310

V. Rod] and M. Schacht

We will verify a similar bound for the second term in (15). For that we will split the second term of (15) into two parts and rewrite each of the parts (see (17) and (18) below). We consider the induced subgraph F* of F, which we obtain by removing the vertices labeled £-1 and £ from F. For a copy F* of F* in G let X£-l (F*) and X£(F*) be those vertex sets such that for every pair X£- l E X£-l (F*) and X£ E X£(F*) of distinct vertices, those two vertices extend F* in G to a copy of F- . More precisely, if Xl , . . . , X£-2 is the vertex set of F* then we set X£-I(F*)

n

=

rC( Xi)

and

X£(F*)

n

=

i: {i,£-I}EE(F)

rC(Xi),

i : {i,£}EE(F)

where rc(x) denotes the set of neighbours of X in G. To simplify the notation, below we will write X£-l or X£ instead of X£-l (F*) or X£(F*) as F* will be clear from the context. Since by definition edges contained in X£-l nx£ are counted twice in e(X£-I,X£) (cf. (2)) we observe for the first part of the second term in (15) that

(17)

( ..II

L Xl ,...,Xt E(V )t

llE(Xi,Xj))llE(X£-I ,X£)

{t,]}EE(F-)

= ~e(X£_I'X£), F*

Moreover, we have for the second part of the seond term in (15) (18)

( .. II

L Xl, ...,XtE(V)t

llE(x i, Xj)) dp(X£-b xd

{t,] }EE(F-)

=

L L

d(Vi, Vj)IX£-1 n ViIIX£ n Vjl

P* i jij E[t]

and , consequently, we can bound the second term in (15) by

(19)

L le(X£-I,X£) - L P*

d(Vi, Vj)IX£-1 n ViIIX£ n Vjll

i¥jE[t]

Finally, we can apply the fact that P is an e-FK-partition in form of (2) and the fact that NF* (G) ::; n£-2 to bound (19) by n£-2 ·6en2. Hence, from (14)-(19) we infer

INF(G) -

L

II

FR {ij,idEE(FR)

wR(ij , ik)

II ij EV (FR)

which concludes the proof of Theorem 18. •

IVi j

II ::; h' + 6e)n£ ::; ,nt,

311

Regularity Lemmas for Graphs

A simple argument based on the principle of inclusion and exclusion yields an induced version of Theorem 16. Let Np{G) denote the number of labeled, induced copies of Fin G. Corollary 17. Let F be a graph with vertex set V{F) = [fl. For every , > 0 there exists c > 0 such that for every G = (V, E) with IVI = nand every c-FK-partition P = (Vi)iE[tj with reduced graph R = Rc{P) we have Np{G) =

L

II

wR{ij,ik)

FR {ij,idEE(FR)

x

II

(l-WR{ij,ik))

{ij,ik}EE(FR)

II IVi I ± ,n

f

j

,

ijEV(FR)

where the sum runs over all labeled copies FR of F in R and FR denotes the complement graph of FR on the same f vertices V{FR) . Proof. Let F be a graph with V{F) = [f] and let K f be the complete graph on the same vertex set. Let e be sufficiently small, so that we can apply Theorem 16 with " = , /2(;)-e(F) for every graph F' ~ K f which contains F. Let G, an c-FK-partition P, and a reduced graph R = Rc{P) be given. Due to the principle of inclusion and exclusion we have Np{G)

=

L

(_l)e(F')-e(F) NF'(G) ,

Fr;F'r;Kl

where we sum over all supergraphs F' of F contained in Theorem 16 for every such F' we obtain

tc'. Applying

Np{G)

=~>_l)e(F')-e(F)(~ . F

.

II , WR{ij,i k) . II IVijl)±,n

F R {tj,tdEE(FR)

f

,

tjEV(FR)

where the outer sum runs over all F' with F ~ F' ~ K f and the inner sum is indexed by all copies F~ of F' in R. We can rewrite the main term by rearranging the sum in the following way: First we sum over all possible labeled copies FR of F in R. Note that this fixes a unique labeled copy

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V. Rodl and M. Schacht

Kl(FR) of K l as well, and in the inner sum we consider all graphs F~ in R "sandwiched" between FR and Kl(FR)' This way we obtain Np(G)

=

L

±,nl

L

(-1

FR FRr;FRr;Kt(FR)

=L II

wR(ij,ik)

L

L

j

ijEV(FR)

II

wR(ij, ik)

{ij ,ik}EE(FR)\E(FR)

wR(ij, ik)

FR {ij,idEE(FR)

x

II IVi I

II IVijl

(_1)e(FR)-e(FR)

II

ik)

ijEV(FR)

FRr;FRr;Kt(FR)

=

W R(ij,

{ij,ik}EE(FR)

FR {ij ,ik}EE(FR)

x

II

)e(FR)-e(FR)

II

IVi I j

ijEV(FR)

II

(I-WR(ij,ik)),

{ij ,ik}EE( Kt(FR)) \E(FR)

which concludes the proof.

_

3.2. The local counting lemma For graphs F and G, a partition P = (Vi)iE[tj of V(G), and a labeled copy FR of Fin R with V(FR) = {il, ... ,ie} we denote by N F ( G[FRJ) the number of partite isomorphic-copies of FR (and hence of F) in G induced on Vi! U... UVi r In other words, N F (G[FRJ) is the number of edge preserving mappings ip from V (FR) to Vi! U. . . UVi t such that
313

Regularity Lemmas for Graphs

Recall, that by definition the edge set E(R) of a reduced graph of a Szemeredi-partition P corresponds to the regular pairs of P. Consequently, for a copy FR of Fin R we require that all edges of FR correspond to regular pairs.

Theorem 18. Let F be a graph with f vertices. Forevery, > 0 there exists E > 0 such that for every G = (V, E) with IVI = n and every e-Szeuietedipartition P = (Vi) iE[tj with reduced graph R = Ra(P, E) we have for every labeled copy FR of F in R with V(FR) = {il, ... ,id

II

NF(G[FRJ) =

wR(ij,ik)

{ij,ik}EE(FR)

II

IVijl

±,

ij EV(FR)

II

IVijl·

ij EV(FR)

Theorem 18 concerns the number of copies of a fixed graph F and will only give interesting bounds if we can assert y «; TI{i j ,idEE(FR) wR(ij ,ik). Moreover, it was shown by Chvatal, Rodl, Szemeredi, and Trotter [7], that if H is a graph of bounded degree with cnl t vertices (for some appropriate c > 0 depending on min{ij,ik}EE(FR) wR(ij ,ik) and D..(H)) and there exists a homomorphism from H into FR, then, under the same assumptions as in Theorem 18, G[FR] contains a copy of H . A far reaching strengthening, the so-called blow-up lemma, was found by Komlos, Sarkozy, and Szemeredi [24] . The blow-up lemma allows, under some slightly more restrictive assumptions, to embed spanning graphs H of bounded degree .

Proof. We prove Theorem 18 by induction on the number of edges of F . Since the theorem is trivial for graphs with no edges and it follows from the definition of e-Szemeredi-partition for E = , for graphs with precisely one edge. Let F be a graph with at least two edges and f vertices. For given, > 0 let E ~ ,/2 be sufficiently small , so that the theorem holds for F- with " = ,/2. Let G = (V, E) be given along with an e-Szemeredi-partition P = (Vi) iE[tj and let FR be a labeled copy of F in R. Without loss of generality we may assume that V(FR) = {I, . . . , f} and that {f -1 , f} is an edge of FR. We denote by Fi the subgraph of FR which we obtain after deleting the edge {f -1, f} from FR. We can express the number of partite isomorphic copies of FR through

N F( G[FRJ)

=

L ... L x IEVl

II

XtEV[ {i,j}EE(FR)

llE(Xi,Xj)

314

V. Rod] and M. Schacht

=

II

L ., . L

(llE(Xi,Xj)

xeEl/l {i,j}EE(Fi)

xIEVl

The last expression can be rewritten as d(Y£-l'

+

Y£) x N F -

(G[FR"D

L ... L ( II xIEVl

xeEVe

llE(Xi, Xj)(llE(X£-l, x£) - d(Y£-l' Y£)) ).

{i ,j}EE(Fi)

From the induction assumption we then infer

II

wR(ij, i k)

{ij ,idEE(FR)

II IVi I ± ~ II IVi I j

ijEV(FR)

j

ijEV(FR)

and , therefore , it suffices to verify

(20)

L'" L ( IT x I EVl

xeEl/l

llE(Xi ,Xj)(llE(X£-l,X£) - d(Y£-l' Y£)))

{i ,j}EE(Fi)

~~

IT IVijl· ij EV (FR )

For that we will appeal to the regularity of P. Let FR be the induced sub graph of FR which one obtains by removing the vertices .e -1 and z. For a partite isomorphic copy F* of FR, let X£-l (F*) ~ Y£-l and X£(F*) ~ lie be those sets of vertices for which any choice of X£-l E X£-l(F*) and X£ E X£(F*) complete F* to a partite isomorphic copy of FR" . (To simplify the notation, below we will write X£-l or X£ instead of X£-l(F*) or X£(F*) as F* will be clear from the context.) Consequently, summing over all partite isomorphic copies F* of F Rin G we obtain

L .. . L ( IT xI E Vl

xeE l/l

{i ,j}EE(Fi)

llE(Xi, Xj) (llE( X£-l, Xi ) - d(lIe-l '

lie)) )

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Regularity Lemmas for Graphs

=

I~e(Xt-I,Xt) -

d(ve-l' ve)IXt-IIIXtll

F*

: :; 2:: le(Xt-I,Xt) - d(ve-l, ve)IXt-IIIXtll P* t-2

: :; II IViI

X

cive-Ilivel,

i= l

where in the last estimate we used the s-regularity of (ve-l, ve) and the obvious upper bound on the number of partite isomorphic copies P* of FHSince e :::; ,/2 the assertion (20) follows and concludes the proof of Theorem 18. • We close this section by noting that an induced version of Theorem 18 can be derived directly from Theorem 18 in a similar way as Corollary 17 (we omit the details). Corollary 19. Let F be a graph with f. vertices. For every, > 0 there exists e > 0 such that for every G = (V, E) with IVI = n and every cSzemeredi-partition P = (Vi)iE[tj with reduced graph R = RG(P, c) the following is true.

For every labeled copy FR of F contained in a clique Kjz ~ R with V(FR) = V(Kk) = {il, . .. , it}

II {ij ,iklEE(FR)

IT

wR(ij,ik)

(l-WR(ij ,ik))

{ij ,i kl E( V (; R »)\E (FR )

= N

p(G[FR]) ±,

IT

II

IVijl

i jEV(FR)

1Vij I,

ijEV(FR)

where N p(G[FR]) denotes the number of labeled, induced, partite isomorphic copies of FR in G[FRJ = G[Vi 1 U...UVieJ. • Note that by assumption of Corollary 19 and the definition of the reduced graph for Szemeredi-partitions we require for FR ~ R with V(FR) = {ill " " ill, that (Vij ' Vik ) is s-regular for every pair {ij, id and not only for pairs corresponding to edges of FR.

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V. Rodl and M. Schacht

3.3. The removal lemma and its generalizations A direct consequence of the local counting lemma is the so-called removal lemma. Answering a question of Brown, S6s, and Erdos [6, 45] Ruzsa and Szemeredi [43] established the triangle removal lemma. They proved that every graph which contains only o(n 3 ) triangles can be made triangle free by removing at most o(n 2 ) edges. This result was generalized by Erdos, Frankl, and Rodl [10] from triangles to arbitrary graphs.

Theorem 20 (Removal lemma for graphs). For every graph F with £ vertices and every ry > 0 there exists c > 0 and no such that every graph G = (V, E) on n ~ no vertices with N F(G) < en£ , there exists a subgraph H = (V, E') such that N F(H) = 0 and IE \ E'I ::; ryn 2 • While the original proof of Ruzsa and Szemeredi was based on an iterated application of the early version of the regularity lemma, Theorem 12, the proof given in [10] is based on Szemeredi 's regularity lemma, Theorem 7. We remark that even in the triangle case both proofs give essentially the same tower-type dependency between c and n, Le., c is a polynomial in liT, where T is a tower of 2's of height polynomial in 1/ry. It is an intriguing open problem to find a proof which gives a better dependency between c and ry.

Proof. Suppose that G = (V, E) is a graph which even after the deletion of any set of at most ryn 2 edges still contains a copy of F . We will show that such a graph G contains at least en" copies of F. For that we apply Szemeredi's regularity lemma, Theorem 7, with

c'}

. {ry c = min 8£2' 3£2

and

to =

1 ry

-,

where e' is given by the local counting lemma applied with F and =

,

~

3

(!!.) e(F) 4 '

and obtain an s-Szemeredi-partition 'P = (Vi)iE [tj of V. Next we delete all edges e E E for which at least one of the following holds :

• e

~

Vi for some i

E

[t],

• e E E(Vi, Vj) for some 1 ::; i < j ::; t such that (Vi, Vj) is not s-regular,

Regularity Lemmas for Graphs

317

• e E E(Vi, \-j) for some 1 ::; i < j ::; t such that d(Vi , \-j) ::; 'T//2. Simple calculations show that we delete at most 'T/n 2 edges in total. Let G' be the graph, which we obtain after the deletion of those edges. Due to the assumption on G, the graph G' must still contain a copy Fo of F. Therefore the reduced graph R = RG{P,E) must contain a copy of a homomorphic image F~ of F for which wR(ij ,jk) ~ 'T//2 for all {ij ,jk} E E(F~). If F~ is a copy of F , then the local counting lemma, Theorem 18, implies that G' contains, for sufficiently large n at least

copies of F. Consequently, NF(G) ~ NF(G') ~ enl, for some e only depending on 'T/ and Ts z ( min { 17/(8£2), E' / (3£2) } , 1h), where E' only depends on F and 'T/. In other words, there exists such a e which only depends on the graph F and 17 as claimed. The case when F~ is not isomorphic to F is very similar. For example, we may subdivide every vertex class Vi into £ classes, Vi,lU ", UVi,l = Vi , and obtain a refinement Q. It follows from the definition of e-regular pair , that if (Vi , \-j) is s-regular, then (Vi,a , \-j,b) is (3£2 E)-regular for any a, b E [£] and d(Vi,a, \-j,b) ~ d(Vi , \-j) - 2£2E. Since F~ was contained in R, the reduced graph S = SG/(Q,3f 2E) must contain a full copy FR of F for which wR(ij ,jk) ~ 'T//2 - 2f2E ~ 'T//4 for all {ij ,jk} E E(FR) and the local counting lemma yields N F( G) 2': cn l for

It was shown by Ruzsa and Szemeredi [43] that the removal lemma for triangles can be used to deduce Szemeredi's theorem on arithmetic progressions for progressions of length 3, which was earlier (and with better quantitative bounds) proved by Roth [42]. This connection was generalized by Frankl and Rodl [11 , 35], who showed that the removal lemma for the complete k-uniform hypergraph with k + 1 vertices implies Szemeredi's theorem for arithmetic progressions of length k + 1. Moreover, Frankl and Rodl [11] verified such a removal lemma for k = 3 (see also [32] for the general removal lemma for 3-uniform hypergraphs) and Rodl and Skokan [40] for k = 4. The general result for k-uniform hypergraphs, based on generalizations of the regularity lemma and the local counting lemma for hypergraphs,

318

V. Rod] and M . Schacht

was obtained independently by Gowers [18] and by Nagle, Skokan, and authors [33, 39, 41]. Moreover Solymosi [44] and Tengan, Tokushige, and authors [38] showed that this result also implies multidimensional versions of Szemeredi's theorem first obtained by Furstenberg and Katznelson [13, 14]. Besides those extensions to hypergraphs, generalizations of Theorem 20 for graphs were proved by several authors. In particular, the regularity lemma of Alon , Fischer, Krivelevich, and M. Szegedy, Theorem 10, was introduced to prove the natural analog of the removal lemma for induced copies of F . In fact, the proof of this statement is already considerably more involved. Later, Alon and Shapira [4, 3] generalized those results by replacing the fixed graph F by a possibly infinite family of graphs F . All those proofs relied on Theorem 10. The most general version , due to Alon and Shapira [3], states the following.

Theorem 21. For every (possibly infinite) family of graphs F and every TJ > 0 there exist constants c > 0, C > 0, and no such that the following holds. Suppose G = (V, E) is a graph on n 2 no vertices. If for every £ = 1, ... , C and every F E F on £ vertices we have N p(G) < cn£, then there exists a graph H = (V, E') on the same vertex set as G such that IELE'I ::; TJn 2 and Np(H) = 0 for every FE F. • The proof of Theorem 21 is more involved and we will not present it here. The hypergraph extensions of Theorem 21 were obtained in [36]. Theorem 21 has interesting consequences in the area of property testing. Roughly speaking, it asserts that every graph G which is "far" (more than TJn 2 edges must be deleted or added) from some given hereditary property A (a property of graphs closed under isomorphism and vert ex removal) must contain "many" (cnIV(F)I) induced copies of some graph F ~ A of fixed size (I V (F) I ::; C). Consequently, a randomized algorithm can easily distinguish between graphs having A and those which are far from A, provided A is decidable. One of the main questions in property testing, posed by Goldreich, Goldwasser, and Ron [16], concerns a natural characterization of properties allowing such a randomized algorithm. With respect to this question, the result of Alon and Shapira shows that all decidable, hereditary properties belong to that class (see [3] and [27] for more details) . An alternative proof of Theorem 21 was found by Lovasz and B. Szegedy [28] . This new proof was based on the limit approach for sequences of dense graphs of those authors [29], which can be viewed as an infinitary iteration of Theorem 1. We will briefly explain this approach in the next section.

Regularity Lemmas for Graphs

319

4. GRAPH LIMITS

In Section 2 we first introduced the (weak) regularity lemma of Frieze and Kannan and from an iterated version we deduced Szemeredi's regularity lemma and the (c, r)-regularity lemma. Iterating Szemeredi's regularity lemma then resulted in the (strong) regularity lemma of Alon, Fischer, Krivelevich, and M. Szegedy, which was the key ingredient for the proof of Theorem 21. It seems natural to further iterate any of those regularity lemmas. In fact , this was studied by Lovasz and B. Szegedy [29]. Roughly speaking, those authors iterated the regularity lemma of Frieze and Kannan infinitely often. Below we will briefly outline some of their ideas. Note that due to the discussion above it does not matter which regularity lemma we iterate infinitely often , since we "pick up the other ones along the way". Suppose (Gi)iEN is an infinite sequence of graphs with IV (Gi ) I -t 00 and (ci)iEN is a sequence of positive reals which tend to O. Now we may apply Theorem 1 with Cl and to = 1 to every sufficiently large graph G, of the sequence. This way we obtain for every such graph G, an cl-FKpartition Pi,l and a reduced graph ~,l = Re i (Pi,l) . Note that all those partitions have at most TFK (cl) parts. Hence, if we discretize the weights of the reduced graphs ~,l by quantities of up to Cl, we note that there TFK ("l »)

are only rl/cll ( 2 different possible reduced graphs. Consequently, there exists a weighted graph Rl with at most TFK(cl) vertices such that ~,l = R; for infinitely many choices i E No In other words, there exists an infinite subsequence (G i j ) jEN such that for every member there exists an cl-FK-partition, which yields R l as the reduced graph. We rename this sequence to (Gt) iEN and let (pl) iEN be the corresponding sequence of cl-FK-partitions. We then repeat the above procedure with C2 for the infinite subsequence where the c2-FK-partitions should refine the cl-FK-partitions. This way we obtain a reduced graph R2, an infinite subsequence (Gn iEN of

(GD iEN'

(Gt) iEN ' and a corresponding sequence of (pl) iEN of c2-FK-partitions. Repeating this step for every Cj with j E N, we obtain a sequence of subsequences (G{) iEN of graphs and a sequence of reduced graphs (Rj) jEN' To avoid sequences of sequences of graphs we may pass to the diagonal sequence and let (Hj)jEN = (G;) jEN which is a subsequence of the original

320

V. Rod] and M. Schacht

sequence of graphs (Gi)iEN ' Moreover, for every j EN and every k = 1, ... ,j let be the ck-FK-partition of Hj with reduced graph Ri:

QJ

Summarizing the above, we have argued that for every infinite sequence of graphs (Gi)iEN with IV(G i)1 -t 00 and every sequence of positive reals (ci)iEN there exists a subsequence (Hj)jEN of (Gi)iEN' and a sequence of reduced graphs (Rj)jEN such that

(a) for every j E Nand k = 1, . .. ,j exist an ck-FK-partition with reduced graph Rk and

(b) for every j EN and k = 1, . . . , j - 1 the partition

QJ of Hj

Q;+l refines QJ.

In some sense the graphs in the sequence (Hj)jEN become more and more similar, since they have almost identical FK-partitions for smaller and smaller c. On the other hand, they may have very different sizes, which makes it hard to compare them directly. In order to circumvent that we may scale them all to the same size, by viewing them as functions on [0,1]2. We will now make this more precise. Let Rj be a reduced graph with tj vertices. We split [0,1] into tj intervals Ij,lU ", Ulj,tj = [0,1] each of size l/tj. We then define the symmetric, stepfunction Rj : [0,1]2 -t [0,1] by setting A

Rj(x ,y) =

{WRj(k'.e), 0,

if (x,y) belongs to the interior of Ij ,k x Ij,e, otherwise .

Recall that those reduced graphs Rj came from refining partitions. It will be important to impose the same structure for the implicit partitions Ij,lU" , Ulj ,tj = [0,1] for j E N. In particular, we recall that every vertex of Rj represents a vertex class of ~, ' for every j' 2: j and those vertex classes were split into tj+!/tj vertex classes in ~;l for every j' > j . In other words, we may view every vertex of Rj of being split into tj+!/tj vertices of R j+! . Therefore , we require that for every j the partition Ij+!,l U... Ulj+!,tj+l refines Ij ,lU... Ulj ,tj in the same way, i.e., the tj+!/tj intervals Ij+l ,x representing vertices of R j + l which came from a given vertex y of Rj form a partition of Ij,y. This way we view the sequence of reduced graphs (Rj)jEN as a sequence

If

-t [0,1]. Similarly, we may view of symmetric step-functions from [0, the sequence of graphs (Hj)jEN as a sequence of symmetric step-functions

321

Regularity Lemmas for Graphs

from [0,1]2 -t {O, I}. Here for a graph H, on nj vertices we split [0,1] into nj intervals Jj,l U. . . UJj,nj = [0, 1] (identified by the vertices of Hj) and we set I , if (x, y) belongs to the interior of Jj ,u x Jj ,v and {u,v} E E(Hj) ,

iIj(x,y) = {

0, otherwise.

Here we require that the labeling of the vertices of Hj is "consistent", i.e., if u is a vertex contained in the k-th vertex class of the fixed cj-FK-partition of n; then we impose that Jj ,u ~ Ij,k' After this embedding we can rewrite the property that Rj is the reduced graph of an cj-FK-partition of H j , by (21)

°

for some €j which tends to as Cj tends to 0, where (here and below) the supremum is taken over all (Lebesgue) measurable subset U ~ [0,1]. (Note that iIj and Rj are piecewise linear and, hence, (Lebesgue) measurable on [0,1]2.) Moreover, we can rephrase the global counting lemma, Theorem 16: Let F be a graph with V(F) = [f] and let j be sufficiently large (so that Cj is sufficiently small). Then (22)

°

where for fixed F we have i'j -t as Cj -t 0. It was proved by Lovasz and B. Szegedy in [29] that, due to property (b) above, the sequence (Rj)jEN converges almost everywhere to a measurable, symmetric function R : [0,1]2 -t [0,1] and that (21) and (22) stay valid in the limit . The function R is called the limit of the sequence (Hj)jEN'

I

Theorem 22. For every sequence of graphs (Gi)iEN with V(Gdl -t 00 there exists a subsequence (Hj)jEN and a sequence of reduced graphs (Rj)jEN' and a measurable, symmetric function R : [0,1]2 -t [0,1] such that

(i) Rj converges pointwise almost everywhere to

R,

322

V. Rod] and M. Schacht

(ii) Iim sup )-+00

I(

U~[O,1] Juxu

Hj(x, y) - R(x,y) dx dyl = 0,

and

(iii) for every fEN and every graph F with V(F) = [f]

• The proof of Theorem 22 indicated above, essentially follows the lines of (b) of Theorem 2.2 in [29] (see Lemma 5.1 the proof of the implication (a) and 5.2 in [29]) . Based on Theorem 22 Lovasz and B. Szegedy [28] gave a different and conceptually simpler proof of Theorem 21. The proof of the generalization of Theorem 21 to k-uniform hypergraphs in [36] followed similar ideas (see also [5]) .

*

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[3] N. Alon and A. Shapira, A characterization of th e (natural) graph properties testable with one-sided error, SIAM J. Gomput., 37 (2008), no. 6,1703-1727.

[4] N. Alon and A. Shapira, Every monotone graph property is testable, SIAM J. Gomput ., 38 (2008), no. 2, 505-522 .

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P. E. Haxell, Y. Kohayakawa and T. Luczak, Turan's ext remal problem in random graphs : forbidding odd cycles, Gombinatorica, 16 (1996), no. 1, 107-122 .

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Y. Kohayakawa, Szemeredi 's regularity lemma for sparse graphs, Foundations of computational mathematics (Rio de Janeiro, 1997), pp . 216-230 .

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J . Koml6s, G. N. Sarkozy and E. Szemeredi, Blow-up lemma, Gombinatorica, 17 (1997), no. 1, 109-123.

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J . Koml6s, A. Shokoufandeh, M. Simonovits and E. Szemeredi, The regularity lemma and its applications in graph theory, Theoretical aspects of computer science (Tehran, 2000), Lecture Notes in Comput. ScL, vol. 2292, Springer, Berlin , 2002, pp .84-112.

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J. Koml6s and M. Simonovits, Szemeredi's regularity lemma and its applications in graph theory, Combinatorics, Paul Erdos is eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., vol. 2, Janos Bolyai Math. Soc., Budapest, 1996, pp. 295-352 .

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L. Lovasz and B. Szegedy, Testing properties of graphs and functions, Israel J. Math., to appear.

[28]

L. Lovasz and B. Szegedy, Graph limits and testing hereditary graph properties, Tech. Report MSR-TR-2005-110, Microsoft Research, 2005.

[29J

L. Lovasz and B. Szegedy, Limits of dense graph sequences, J. Gombin. Theory Ser. B, 96 (2006), no. 6, 933-957.

[30J

L. Lovasz and B. Szegedy, Szemeredi 's lemma for the analyst, Geom. Funct. Anal. , 17 (2007), no. 1, 252-270.

[31]

T . Luczak, Randomness and regularity, International Congress of Mathematicians. Vol. III , Eur . Math. Soc., Zurich, 2006, pp . 899-909 .

[32]

B. Nagle and V. Rodl , Regularity properties for triple systems, Random Structures Algorithms, 23 (2003), no. 3, 264-332 .

[33J

B. Nagle, V. Rodl and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, 28 (2006), no. 2, 113-179.

[34J V. Rodl, unpublished. [35J

V. Rodl, Some developments in Ramsey theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp . 1455-1466.

[36J

V. Rodl and M. Schacht , Generalizations of th e removal lemma, Combinatorica, 29 (2009), no. 4, 467-501.

[37J

V. Rodl and M. Schacht , Regular partitions of hypergraphs: regularity lemmas , Gombin. Probab. Gomput., 16 (2007), no. 6, 833-885 .

[38J

V. Rodl, M. Schacht, E. Tengan and N. Tokushige, Density theorems and extremal hypergraph problems, Israel J. Math., 152 (2006), 371-380.

[39]

V. Rodl and J . Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, 25 (2004), no. 1, 1-42.

[40J

V. Rodl and J. Skokan, Counting subgraphs in quasi-random 4-uniform hypergraphs, Random Structures Algorithms, 26 (2005), no. 1-2, 160-203.

[41J

V. Rodl and J . Skokan, Applications of the regularity lemma for uniform hypergraphs, Random Stru ctures Algorithms, 28 (2006), no. 2, 180-194.

[42J K. F. Roth, On certain sets of integers, J. London Math. Soc., 28 (1953), 104-109. [43J I. Z. Ruzsa and E. Szemeredi, Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. Janos Bolyai, vol. 18, North-Holland, Amsterdam, 1978, pp . 939-945 . [44J

J . Solymosi, A note on a question of Erdos and Graham, Gombin. Probab. Comput., 13 (2004), no. 2, 263-267.

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[45] V. T . 86s, P. Erdos and W. G. Brown , On the existence of triangulated spheres in 3-graphs, and related problems , Period. Math. Hungar., 3 (1973), no. 3-4, 221-228 . [46] E. Szemeredi, On graphs containing no complete subgraph with 4 vertices, Mat. Lapok, 23 (1972), 113-116 (1973). [47] E. Szemeredi, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27 (1975), 199-245, Collection of articles in memory of Juri! Vladimirovic Linnik . [48] E. Szemeredi , Regular partitions of graphs, Problemes combinatoires et theorie des graphes (Colloq. Internat. CNRS , Univ. Orsay, Orsay, 1976), Colloq. Internat. CNR8, vol. 260, CNR8 , Paris, 1978, pp. 399-40l. [49]

T. Tao, Szemeredi 's regularity lemma revisited, Contrib. Discrete Math., 1 (2006), no. 1, 8-28 (electronic) .

Vojtech Rodl

Mathias Schacht

Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, U.S.A .

Institut fur Informatik, Humboldt-Universitiit zu Berlin, Unter den Linden 6, D-l0099 Berlin, Germany

e-mail :

e-mail :

rodl~athcs.emory.edu

schacht~informatik.hu-berlin.de

SOlVAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 327-336.

EDGE COLORING MODELS AS SINGULAR VERTEX COLORING MODELS

BALAzs SZEGEDY Dedicated to the 60th birthday of Laszlo Loiuisz In this paper we study an interesting class of graph paramet ers. These paramet ers arise both from edge coloring models and from limit s of vertex coloring models with a fixed numb er of vertices. Their vertex connect ion matrices have exponent ially bounded rank growth.

1. INTRODUCTION

Vertex coloring models (resp. edge coloring models) were introduced by Freedman, Lovasz and Schrijver. The main idea is that any graph G can be imagined as a particle system where the particles are represented by the vertices (resp. edges). The particles can have a finite number of states (called colors) and the states interact with each other along the edges (resp. vertices). In a vertex coloring model every state and interaction between two states has a weight. This data can be best represented as a weighted graph H on the color set. The partition function of the vertex coloring model is defined as the homomorphism number of G into this weighted graph H (see [1]) . Freedman, Lovasz and Schrijver in [1] give a characterization of the partition functions of vertex coloring models (with real weights) in terms of the so called connection matrices M(p , k) that can be defined for an arbitrary graph parameter p and number k. They show that if rk ( M (p, k) ) grows at most expon entially with k and M (p, k) is positive semidefinite for

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every k then there is a weighted graph H (with positive vertex weights and real edge weights) such that p is the homomorphism function into H. A similar characterization of the partition functions of edge coloring models was obtained by the author [2]. The main motivation of this paper is an interesting observation of L. Lovasz. Let p(G) denote the number of perfect matchings in G. Lovasz noticed that the parameter p is similar to a homomorphism number in the sense that rk ( M (p, k)) has exponential rank growth 2k but M (p, k) is not positive semidefinite. However it is not the homomorphism number into any weighted graph. Note that p is the partition function of an edge coloring model. Motivated by this observation the author found the following curious construction. Let H, be the weighted graph on two vertices with edge weights

and vertex weights

(!t '

-1) . t

Then

p(G) = lim hom (G,H t ) t.-+O

for every G. In particular this formula explains the exponential rank growth because M (p, k) is the limit of matrices with rank 2k . The conclusion of the present paper is that a wide class of graph parameters that are coming from edge coloring models can be represented in a similar way. The generalized vertex coloring models will be called singular vertex coloring models. The main theorem says that if an edge coloring model satisfies a certain finiteness condition (which seems to be satisfied by most reasonable parameters) then its partition function can be represented by a singular vertex coloring model. Note that this result is complemented by a result in [2] saying that partition functions of vertex coloring models can be represented by partition functions of edge coloring models with complex values. Acknowledgement. I am very grateful to Laszlo Lovasz for mentoring me throughout my post-doc years (2003-2005) when these results were obtained.

Edge Coloring Models as Singular Vertex Coloring Models

329

2. FINITE RANK FUNCTIONS ON POLYNOMIAL RINGS We will need some basic facts from commutative algebra. Let R = qXl , X2 , " " x n ] be the polynomial ring in n variables and let F : R -t <e be a <e-linear function . We define the symmetric bilinear form b : R x R -t <e by b(p, q) = F(pq). Let Ib denote the set of polynomials p such that b(p, q) = 0 for every q E R. It is clear that h is an ideal in R. We say that F has finite rank if dim (R/lb) < 00 . In general dim (R/lb) will be called the rank of F . We say that F is semisimple if h is a finite co-dimensional radical ideal in R. A function F is semisimple if and only if R/lb == <e k for some natural number k. Finite rank functions are forming a subspace of the dual space of Rand semisimple functions are forming a subspace of all finite rank functions . The next classical lemma ([3], [4]) provides generating sets for these spaces. Lemma 2.1 (Generating finite rank functions). Let m be a multiset of the variables Xl , X2 , ••• ,Xn and v E <en be a vector. Then the functions F m,v defined by Fm,v(p) = 8mP(v) are all finite rank functions and they linearly span the space of finite rank functions. Furthermore the space of semisimple functions is generated by the functions of the form F v (p) = p(v ). The following lemma is a corollary of the previous one. Lemma 2.2 (Approximation of finite rank functions). Let F be an arbitrary finite rank function on R. Then there are two lists of vectors {Vi} 1=1 and {wd1=1 with Vi , Wi E <en and coefficients {ci/t r i }1=1 with Ci E <e and r, EN such that

for every polynomial R. Proof. By Lemma 2.1 it is enough to prove the lemma for functions of the for F m ,u- Let ei denote the vector in <en whose i-th coordinate is 1 and all other coordinates are O. Let furthermore m; denote the multiplicity of Xi in m for 1 ::; i ::; n. Then it is classical that

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where b = L:~=l ttu: •

3.

EDGE COLORING MODELS OF FINITE RANK

The so-called edge coloring models were introduced by Freedman, Lovasz and Schrijver. In this chapter we introduce a special class of edge coloring models with interesting properties. An edge coloring model is represented by a function f : Nn -+ C. Let C = {CI, C2, . • • , en} be a set whose elements are called colors. By abusing the notation, for a multiset 8 of colors we define f(8) as the evaluation of f on the vector (aI, a2, . .. ,an) where ai denotes the multiplicity of c, in 8 . The partition function Pf of the edge coloring model represented by f is a graph parameter defined by the following formula.

Pf(G) =

L

IT

'l/J : E(G)-+C

vEV (G)

f( 8(v , VJ))

where 8( v, VJ) denotes the multiset of values of VJ on the edges adjacent to v. Let us introduce an infinite matrix M] whose rows and columns are indexed by the elements of Nn and M f ( v , w) = f (v +w) for every v , w E Nn. The edge coloring model corresponding to f is called finite rank edge coloring model if the rank of M] is finite. The main method in this paper relies on the fact that an edge coloring model f : Nn -+ C can be turned into a linear function F : q XI , X 2 , . . . , x n ] -+ C in the following way. We define F( X~lX~2 .. . x~n) as f(al , a2 ," " an) and then we extend it linearly to all polynomials. We call F the polynomial extension of f . The main observation is that f is of finite rank if and only if F is of finite rank. This can be seen from the fact that the semigroup Nn is naturally represented as the multiplicative semigroup of monomials in qXI, X 2, ... ,xn ]. We will say that f is a semisimple edge coloring model if the corresponding function F is semisimple.

331

Edge Coloring Models as Singular Vertex Coloring Models

4.

SINGULAR VERTEX COLORING MODELS

Similarly to edge coloring models, the so called vertex coloring models were first introduced by Freedman , Lovasz and Schrijver in the paper [1] . We start with the definition of vertex coloring models and connection matrices and then we introduce a generalization of vertex coloring models. Let H be a complete graph on the vertex set [n] = {I, 2, .. . ,n} with weights Q;i on the vertices and weights f3i,j on the edges. The homomorphism number hom (G, H) for a simple graph G is defined as follows. hom (G,H) =

II

L

Q;4>(v)

4> :V(G)--+[n] vEV(G)

II

f34>(v) ,4>(w)'

(v,W)EE(G)

If H is fixed then the graph parameter t H (G) = hom (G, H) is called the partition function of the vertex coloring model corresponding to H. Let 9k denote the set of simple graphs with k vertices labeled by the set [k]. We denote by 9 = 90 the set of graphs in which no vertices are labeled. The set 9k has a commutative semigroup structure with respect to the natural gluing operation along the labeled edges. For two elements a, b E 9k their on product in this semigroup is denoted byab. Let p : 9 -t C be an arbitrary graph parameter. The k-th connection matrix of p is an infinite matrix M(p, k) of the form C~lkxgk such that M(p , k)(a, b) = p(ab) . Freedman, Lovasz and Schrijver observed that if H is a fixed weighted graph on n vertices then rk(M(tH,k)) ~ n k .

In other words, the rank growth of the connection matrices is at most exponential. Note that the above formula for k = 0 implies that tn is multiplicative when taking disjoint unions of graphs. Now we are ready to define the generalization of vertex models. Let H be as above with the modification that the edge weights are in C[t] and the vertex weights are of the form eft" where e E C and sEN. We say that H gives rise to a singular vertex coloring model if

p(G) = lim hom (G, H) t--+O

EC

for every G E 9. The graph parameter p is called the partition function of the singular vertex model. Every vertex coloring model is a singular vertex coloring model in a way that the edge and vertex weights don't depend on t. The following lemma follows trivially from the fact that M(p, k) = IimHO M(tH, k) and that rk (M(tH' k)) ~ n k •

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B. Szegedy

Lemma 4.1 (Exponential rank growth). Let H be a singular vertex coloring model. Then its partition function P satisfies rk ( M (p, k)) S; n k for every kEN where n is the number of vertices in H. Th e main theorem in this paper is the following. Theorem 1 (Finite rank edge models as singular vertex models). The partition function of any finite rank edge coloring model is the same as the partition function of some singular vertex coloring model. The next curious corollary says that the vertex connection matrices of partition functions of finite rank edge coloring models have exponential rank grows. Note that the edge connection matrices of edge coloring models have exponential rank grow. Corollary 4.1 (Exponential rank growth II). The partition function pf of any finite rank edge coloring model satisfies rk ( M (t f ' k)) S; ck for every kEN and constant c depending on f . For two vectors v = (aI, a2, . . . , an) and w = (bl , b2 , . . . , bn) in en let (v, w) denote the sum alb l + a2b2 + ...+ anbn . For proving theorem 1 we will make use of the next lemma. Lemma 4.2 (Semisimple edge models as vertex models). Let be a semisimple edge coloring model of the form

f : Nn -+ e

d

F(p) = LAiP(Vi) i= l

where Ai E e and Vi E en . Then the partition function of f is the same as the homomorphism number into the weighted graph H on the vertex set [d] with weights ai = Ai, f3i,j = (Vi,Vj) 1 S; i ,j S; d. Proof. Let Ii : Nn -+ e denote the function whose value on (aI, a2, . .. , an) is the substitution of Vi into the monomial X~l X~2 ••. , x~n. We have that f = 2:1=1 Aili ' Let Ji : C -+ C denote the function whose value on Cj is the j - th component of Vi. Then

Pf(G) =

L

d

II L Aili ( S(v,1/1))

1{! :E(G)-+C vEV(G) i=l

=

L

L

II

1{! :E(G)-+C 4> :V(G)-+[d) vEV(G)

A4>(v)f4>(v) (S(v, 1/1))

333

Edge Coloring Models as Singular Vertex Coloring Models

~

~ (II

¢ :V(G)--+[d) 'l/J :E(G)--+C

II

x

J¢(u)

A¢(V))

vEV(G)

('I/J(e)) J¢(w) ('I/J(e))

e=(u,w)EE(G)

=

~

¢ :V(G)--+[d)

(II vEV(G)

A¢(V))

II

(V¢(u),U¢(w)) .

•

(u,W)EE(G)

Now we are ready to prove the main theorem. Proof of Theorem 1: Let f be a finite rank edge coloring model and F be its polynomial extension. Lemma 2.2 yields a one parameter family of semisimple functions Ft such that limHo Ft(q) = F(q) for every n variable polynomial q. Let f(t) be the edge coloring model obtained form F t by restricting it to monomials. We have that limHo f(t)(al ' a2, . . . , an) = f(aI,a2, ... ,an) for every (al ,a2, .. . , an ) E en and so limHoPj(t)(G) = Pj(g) for every simple graph G. Now using Lemma 4.2 and formula (1) the proof is complete.

5. HYBRID MODELS AND DIRECTED EDGE MODELS In this chapter we discuss a common generalization of vertex and edge coloring models that were first introduced by M. Freedman. We call them hybrid models. A half edge in a graph G is an incident pair (v,e) of a vertex v and edge e. We denote by E*(G) the set of half edges in G. Roughly speaking, in a hybrid model a graph is imagined as a particle system in which the half edges can have different states (or colors) that interact with each other both at the vertices and along the edges. The precise definition is the following. Assume that beside a function f : Nn -+ e a symmetric matrix A E enxn is given whose rows and columns are indexed by the elements of the color set C. The pair (I, A) is called a hybrid model. Similarly to edge coloring models, we say that (f, A) is a finite rank hybrid model if the

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B. Szegedy

polynomial extension F of f has finite rank. The partition function p of the hybrid model (j, A) is given by

p(G) =

L (IT

f(S(V,7jJ)))

1jJ :E*(G)--+C VEV(G)

IT

A(7jJ(v,e),7jJ(w,e)) .

e=( v,w)EE(G)

where S(v ,7jJ) denotes the multiset of the values of 7jJ on the half edges incident to v.

It can be derived from the results in [2] that the partition function of any hybrid model is always equal to the partition function of some edge coloring model. So in the sense of partition functions they are not more general. However some combinarial parameters arise more naturally from hybrid models than from pure edge coloring models . For example a variant of the edge coloring model in which edges have colors and directions can be represented as hybrid models . An important feature of hybrid models is that they have finite rank if and only if the representing edge coloring model has finite rank. This allows one to use the machinery developed in this paper for hybrid models .

6.

EXAMPLES AND REMARKS

Finite rank edge coloring models are very common in the sense that edge coloring models with "nice" combinatorial meaning are typically of this type. We list a few conditions that give a lot of finite rank models. Let I, g : Nn -+ C be edge coloring models .

1.) If f takes only finitely many non 0 values then it has finite rank. 2.) If f and g have finite rank then cf, f + g and f g have finite rank where c E C and the operations are performed pointwise on Nn . Together with the fact that constant functions have finite rank, it implies that any polynomial expression of finite rank functions is again of finite rank. 3.) If f has finite rank and h : Nn+m -+ C is obtained from f in the way that h(al ' a2, .. . , a n+m ) = f(al, a2, . . . , an) then h has finite rank. 4.) If g is obtained form f by permuting the coordinates in Nn and finite rank then g has finite rank.

f

has

Edge Coloring Models as Singular Vertex Coloring Models

335

5.) The coordinate functions Ci(a1, a2, . . . , an) = ai have all finite rank.

6.) Exponentials of the coordinate functions all finite rank.

lei

with a fixed lEe have

Now here are some examples:

N umber of perfect matchings. Let f be the two variable edge coloring model with f(a , b) = 1 if a = 1 and 0 if a i= 1. Conditions 1. and 3. show that f has finite rank. The corresponding partition function is the number of perfect matchings. The polynomial extension F of f is equal to F X 1 ,(O,1) ' Thus the semisimple approximation of F given by Lemma 2.2 is Ft (q) = q(t, 1) - q(0, 1)). This implies by Lemma 4.2 that the singular vertex coloring model has two vertices and furthermore a1 = l/t, 2 a2 = -1/t, /31,1 = t + 1, /31 ,2 = /32,1 = /32,2 = 1.

t(

Being three regular. Let f be the one variable edge coloring model with f(a) = 1 of a = 3 and f(a) = 0 if a i= 3. This is of finite rank by condition 1. The partition function of f gives 1 if and only if Gis 3-regular. In this case Ft(q) = b(q(3t)-3q(2t)+3q(t)-q(0)) and so the singular vertex coloring model has four points. The vertex weights are -1/6t3 , 3/6t 3 , -3/6t3 , 1/6t 3 and the edge weights are /3i ,j = (i - 1) (j - 1)t 2 .

N umber of Eulerian subgraphs. Let f be the two variable edge coloring model with f(a, b) = 1 if a is even and 0 if a is odd. Conditions 2., 3. and 6. together show that f is of finite rank. An interesting fact is that f is semisimple so it can be represented by a normal vertex coloring model (this was first observed by L. Lovasz). Further details are left to the reader. Other examples. Many examples for edge coloring models listed in [2] are of finite rank. This includes the number of fully packed loop configurations, the number of matchings, the number of d-regular subgraphs, the number of proper edge colorings with d-colors and the permanent of the adjacency matrix. We finish the paper with an open question . Question. Is there any graph parameter p with exponentially bounded rank growth (rk ( M (p, k)) ::; £1') which is not the partition function of a finite rank edge coloring model?

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REFERENCES [IJ M. Freedman, L. LOVllsZ and A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs, J. Amer. Math. Soc., 20 (2007), no. 1,37-51.

[2] B. Szegedy, Edge coloring models and reflection positivity, J. Amer. Math. Soc., 20 (2007), no. 4, 969-988.

[3] W. Grabner, Algebraische Geometrie II, Bibliographisches Institut, Mannheim, 1970. [4J 1. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.

Balazs Szegedy University of Toronto Dept. of Mathematics 40 St George St. Toronto, ON M5S 2E4, Canada e-mail:

[email protected]

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

Fete of Combinatorics, pp. 337-353 .

LIST TOTAL WEIGHTING OF GRAPHS

TSAI-LIEN WONG*, DAQING YANGt and XUDING ZHUt

A graph G = (V, E) is (k, k')-total weight choosable if the following is true: For any (k, k')-totallist assignment L that assigns to each vertex v a set L(v) of k real numb ers as permissible weights, and assigns to each edge e a set L( e) of k' real numbers as permissible weights, there is a prop er L-total weighting, i.e., a mapping f : VUE --+ lR such that f(y) E L(y) for each y E VUE , and for any two adjacent vertices u and v, I: eEE(u) f(e)+ f(u) =1= I: eEE( v) f( e)+ f(v) . This paper introduces a method, th e max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of th e form Kn ,m ,l ,l ,.. ., l are (2,2)-total weight choosable and complete bipartite graphs other than K2 are (1,2)-total weight choosable .

1.

INTRODUCTION

Suppose G = (V, E) is a graph. For a vertex v of G, let E(v) be the set of edges of G incident to v, and let E* (v) = E (v) U {v} . An edge weighting of G is a mapping that assigns to each edge e of G a real number f (e). An edge weighting f induces a vertex colouring 'Pf : V -+ lR of G, defined as 'Pf(v) = L:eEE(v) f( e). We say f is a proper edge weighting if the induced vertex colouring 'Pf is proper, i.e., for any edge uv of G, 'Pf(u) =1= 'Pf(v) . The study of edge weighting was initiated by Karoriski, Luczak and Thomason [9] . They made an interesting conjecture: Every connected graphs G =1= K 2 has a proper edge weighting f such that f(e) E {l , 2, 3} for every edge e. This conjecture is still open. It was shown in [3] that the conjecture would "Grant number: NSC97-2115-M-ll0-004-MY2. tSupported in part by NSFC und er grant 10771035 and 10931003 grant SX2006-42 of colleges of Fujian. 'Grant number: NSC97-2115-M-ll0-008-MY3.

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T.-L. Wong, D. Yang and X. Zhu

be true if the set {I, 2, 3} is replaced by {I , 2, . . . , 30}. The result was improved in [2], where it was shown that the conjecture would be true if the set is {I, 2, ... , 16}, and recently, it is shown in [10] that the conjecture would hold if the set is {I, 2, 3, 4, 5}. Bartnicki, Grytczuk and Niwczyk [7] considered the choosability version of edge weighting. A graph is said to be k-edge weight choosable if the following is true: For any list assignment L which assigns to each edge e a set L( e) of k real numbers, G has a proper edge weighting f such that f(e) E L(e) for each edge e. They conjectured that every graph without isolated edges is 3-edge weight choosable, and verified the conjecture for complete graphs, complete bipartite graphs and some other graphs. A total weighting of G is a mapping f : VUE -+ JR which assigns to each vertex and each edge a real number as its weight. For a total weighting i, let
However, it is still unknown if there are constants k, k' such that every graph is (k, k')-choosable. It was shown in [13] that complete graphs, trees,

List Total Weighting of Graphs

339

cycles, generalized theta graphs are (2, 2)-choosable, and complete bipartite graphs K 2 ,n are (1,2)-choosable and K 3 ,n are (2,2)-choosable. The method used in [7] and in [13] for the study of edge weight choosability or total weight choosability of graphs is algebraic (see Section 5), which proves the existence of a proper edge weighting or proper total weighting without actually constructing such a weighting. In this paper, we introduce a method that constructs explicitly a proper total weighting of given graphs. Then we apply this method and prove that complete multipartite graphs of the form Kn,m,l,l,...,l are (2,2)-choosable and complete bipartite graphs other than K 2 are (1, 2)-choosable.

2. THE MAX-MIN WEIGHTING METHOD

Suppose L is a (k , k')-total list assignment of a graph G. The max-min weighting method finds a proper L-weighting of G step by step. At each step, we find a vertex v and determine the weights of all the elements of E*(v). Let fa : E*(v) -+ lR be a map such that for each Y E E*(v), fo(Y) E L(y). We call fa a partial L-weighting of G for vertex v . If fa is a partial L-weighting of G for v , let i.pfo(v) = L:yEE*(v) fo(Y). Then for any total weighting f of G which coincides with fa on E*(v), we have i.pf(v) = i.pfo(v) . I.e., the colour of v is determined by the partial L-weighting fa· Given a partial L-weighting fa of G for vertex v. We define a (k, k')-total list assignment L l for G l = G - v as follows:

L(Y)' if Y is a vertex not adjacent to v or Y is an edge of G l, Ll(y)=

{

{w+fo(e): WEL(y)}, if y is a vertex adjacent to v and e = yv.

We call L, the (k , k')-totallist assignment induced by L and fa. Suppose L, is the (k, k')-totallist assignment of G l induced by Land fa. If h is an Ll-weighting of G l, then we combine h and fa to obtain a

340

T.-L. Wong, D. Yang and X. Zhu

weighting

1 of G as follows : lI(y) - lo(e), if y is a vertex adjacent to v and e = yv,

I(Y) =

lI(y), if y is a vertex not adjacent to v or y is an edge of G l , 10(Y), if y

E

E*(v)

We call 1 the L-weighting of G induced by II and 10. The following lemma is obvious and its proof is omitted.

Lemma 2.1. Suppose L is a (k, k')-totallist assignment of a graph G, 10 is a partial L-weighting of G for vertex v, and L l is the (k, k')-total list assignment of G l = G - v induced by Land 10 . If II is an Ll-weighting of G l , and 1 is the L-weighting of G induced by II and 10, then for each u E V(Gd, 'Pf(u) = 'Ph(u) and 'Pf(v) = 'Pfo(v) . • For each y E VUE, let Lmax(y) = maxL(y) be the maximum number in L(y), and let Lmin(Y) = minL(y) be the minimum number in L(y) . Note that if L(y)1 = 1, then Lmax(y) = Lmin(y). Otherwise, Lmax(y) > Lmin(Y) .

I

Let

'PL,max(V) =

L

Lmax(y)

YEE*(v)

'PL,min(V) =

L

Lmin(y) .

YEE*(v)

It follows from the definition that 'PL,max(V) = max'Pf(v) and 'PL,min(V) = min'Pf(v), where the maximum and minimum is taken over all L-weightings 1 of G. So for any L-weighting 1 of G, we have

For a vertex v of G, Nc(v) = {u : u is a neighbour of v} , and Nc[v] = Nc(v) U {v} . A vertex v of G is locally maximum (with respect to L) iffor any u E Nc(v), 'PL,max(U) ~ 'PL,max(V), and v is locally minimum (with respect to L) if for any u E Nc(v), 'PL,min(U) ~ 'PL,min(V). A partial L-weighting of G for a vertex v is said to max-weights v if I(y) = Lmax(y) for each y E E*(v) . A partial L-weighting of G for a vertex

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v is said to min-weights v if f(y) = Lmin(y) to for each y E E*(v). If f is a partial L-weighting of G that max-weights v, then
= {

U E Nc(v) : epL,max(U)

=

epL,max(V)}.

To extend fo to a proper L-weighting f of G, we need to make sure that no vertex U E QL,max(V) will be "coloured" with epL,max(U). For this purpose, it suffices that some y E E*(u) is assigned a weight smaller than Lmax(y). I.e., U is not max-weighted (under the list L). Similarly, assume v is a locally minimum vertex , and fo is a partial L-weighting of G for v that min-weights v , and QL,min(V)

= {U E Nc(v) : epL,min(U) = epL,min(V)} .

To extend fo to a proper L-weighting f of G, we need to make sure that U E QL,min(V) is not min-weighted (under the list L) . This motivates the following definition. Definition 2.2. Suppose G = (V, E) is a graph and L is a (k , k')-total list assignment of G. Let So,To be two subsets of V (each of So,To may be empty and So,To need not be disjoint). A proper (L ;So,To) -weighting of G is a proper L-weighting f of G such that for each v E So, epf(v) i= epL,max(V) and for each v E To , epf(v) i= epL,min(V).

The following theorem contains the key idea of max-min weighting method. Theorem 2.3. Suppose G = (V, E) is a graph, k' 2: 2 and L is a (k, k')-total list assignment of G, and So, To are two subsets of V .

• Assume v is a locally maximum vertex and v tf. So. In case k = 1, we further assume that v is not an isolated vertex. Let fo be the partial

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L-weighting of G that max-weights v. Let G1 = G - v, and let L 1 be the (k, k')-totallist assignment induced by Land fo. Let 51 = So U QL,max(V), T 1 = To - Nc[v].

If h is a proper (L 1; 51,T1)-weighting of G1, then the L-weighting induced by hand fo is a proper (L; So, To)-weighting of G .

• Assume v is a locally minimum vertex and v ~ To. In case k = 1, we further assume that v is not an isolated vertex. Let fo be the partial L-weighting of G that min-weights v. Let G1 = G - v, and let L 1 be the (k , k')-totallist assignment induced by Land fo. Let 51 = So - Nc[v],

T1 = To U QL,min(V) .

If h is a proper (L 1; 5l, T1) -weighting of G1, then the L-weighting induced by hand fo is a proper (L; So, To)-weighting of G. Proof. Assume v is a locally maximum vertex and v ~ So. Let fo be the partial L-weighting of G that max-weights v. Let G1 , 51 , T 1 be defined as above. Assume h is a proper (L 1 ; 51, T1)-weighting of G1 . Let 9 be the L-weighting of G induced by hand fo. To prove that 9 is a proper (L; So, To)-weighting of G, it suffices to show the following: 1. For any U E QL,max(V) , 'Pg(u)

=1=

'PL,max(U),

2. For any U E So, 'Pg(u) =1= 'PL,max(U). 3. For any U E To, 'Pg(u)

=1=

'PL,min(U),

By Lemma 2.1, 'Pg(u) = 'Ph(u) for U =1= v and 'Pg(v) = 'PL,max(V). Since QL,max(V) ~ 51 and h is a proper (L l ; 51, Tl)-weighting of G, it follows that for U E QL,max(V) ~ 51 , 'Pg(u) = 'Ph(u) =1= 'PL,max(U), So (1) is satisfied. Since So ~ 51, we also have 'Pg(u) = 'Ph(u) =1= 'PL,max(U) for every U E So. So (2) is satisfied. Assume U E To. If U E T 1, then 'Pg(u) = 'Ph(u) > 'PL1 ,min(U) = 'PL,min(U), Ifu E Nc(v), then since the edge e = uv is assigned the weight g(e) = fo(e) = Lmax(e) > Lmin(e) (here we use the assumption that k' ~ 2), we conclude that 'PL1 ,min(U) > 'PL,min(U), Therefore 'Pg(u) ~ 'PL1 ,min(U) > 'PL,min(U). If U = v, then since either k ~ 2 or v is not isolated , we have 'PL,max(V) > 'PL,min(V) (here again we use the assumption that k' ~ 2). Therefore 'Pg(v) = 'PL,max(V) =1= 'PL,min(V). Thus (3) is satisfied . The case that v is locally minimum is symmetric, and the proof is the same, except that max is replaced by min, QL,max ( v) is replaced by QL,min(V) , and So, To are interchanged, and 51, Ti are interchanged. -

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3.

ApPLICATION OF THE MAX-MIN METHOD

In this section, we use the max-min weighting method to prove that complete multipartite graphs of the form K n ,m ,I ,I ,...,1 are (2,2)-choosable and complete bipartite graphs other than K 2 are (1, 2)-choosable. In most of the applications of the max-min weighting method in this paper, we max-weight globally maximum vertices and min-weight globally minimum vertices. For a (k, k')-totallist assignment L of G, let VL,max(G) be the set of globally maximum vertices of G and let VL ,min(G) be the set of globally minimum vertices of G. To be precise,

VL ,max(G) = {v VL,min(G) =

E

V(G) : 'Vu

E

V(G), ipL,max(V) ~ ipL,max(U)}

{v E V(G) : 'Vu E V(G),

ipL,min(V) ::; ipL ,min(U)} ,

First we prove that complete multipartite graphs of the form K n ,m ,I ,I ,...,1 are (2,2)-choosable.

Theorem 3.1. Suppose n , m , k ~ 0 are integers. Let G = K n ,m ,I ,I ,...,1 be the complete multipartite graph with one partite set of size n, one partite set of size m and k partite sets of size 1. Denote by A the partite set of size n , and by B the partite set of size m and by C the union of the k partite sets each of size 1. Suppose L is a (2,2)-total list assignment L of G, and So, To are subsets of V(G) such that one of So, To is empty and the other has an empty intersection with A or B . Then there is a proper (L; So, To)-weighting of G. Hence G is (2,2)-choosable. Proof. We prove this theorem by induction on IV(G)I . If IV(G)I = 1, then the conclusion is obviously true. Assume IV (G) I > 1. Without loss of generality, we assume that So = 0 and assume that To n A = 0. • If VL,max(G)

n A i= 0, then let

v E VL,max(G)

n A.

• IfVL,max(G)nA = 0 and VL,max(G)nC i= 0, then let v • If VL,max(G)

n (A U C)

=

0, then let v

E

E

VL ,max(G)nC.

VL ,max(G) n B.

Let fo be the partial L-weighting of G that max-weights v. Let G' = G - v and L, be the (2,2)-total list assignment induced by Land fo. Let SI = So U QL,max(V) = QL,max(V) and let T l = To - Nc[v]. It is easy to verify that if v E A, then T l = 0 and SI n A = 0. If v E C, then T l = 0 and

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8 1 nA = 0. If V E B, then 8 1 = 0 and Tl nA = 0. By induction hypothesis, G' has a proper (L l ; 8 1 , Tt}-weighting. By Theorem 2.3, G has a proper (L; 80, To)-weighting. • It was proved in [13] that complete graphs are (2,2)-choosable. The proof in [13] uses Combinatorial Nullstellensatz and does not provide an algorithm that constructs a proper L-total weighting of K n for a given (2,2)total list assignment L of K n . The proof above gives a linear time algorithm that constructs a required proper L-total weighting of K n ,m ,I ,I ,...,I ' In [13], the problem whether complete bipartite graphs are (2,2)-choosable was studied. It was proved there that for any positive integer n, K2,n are (1, 2)-choosable and K 3 ,n are (2, 2)-choosable. But the question whether all complete bipartite graphs are (2,2)-choosable remained open. Theorem 3.1 answers this question in the affirmative. Theorem 3.2 below shows that complete bipartite graphs are actually (1,2)-choosable, provided that it is not K 2 •

Theorem 3.2. Suppose G = Kn,m i= K 2 is a complete bipartite graph and L is a (1, 2)-totallist assignment of G. If both 80, To are empty sets, then G has a proper (L; 80, To)-weighting. Proof. Assume the theorem is not true and G = Kn,m is a counterexample with minimum number of vertices. Let A, B be the two partite sets with IAI = nand IBI = m. Claim 1. G tJ: {K l ,2, Kl ,3, K2,2, K2,3}. The graphs excluded in this claim are small graphs. One way to prove this claim is to do a case by case check, which seems to be tedious. Another method is to use Combinatorial Nullstellensatz, which will be discussed in Section 5.

Claim 2. None of VL ,max(G) and VL,min(G) is an independent set of G. Proof. Assume to the contrary that VL ,max(G) is an independent set of G, i.e., VL ,max(G) is a subset of A. Let v E VL,max(G) and let fo be the partial L-weighting of G that maxweights v. Let L, be the (1, 2)-totallist assignment of G l = G - v induced by Land fo. Since 80 = To = 0, and QL,max(V) = 0, we have 8 1 = T l = 0 for the sets 8 1 , T l defined in Theorem 2.3. Since G l i= K2, by induction hypothesis, G, has a proper (Ll; 81, Tl)-weighting [i - By Theorem 2.3, the L-weighting f of G induced by iI and fo is a proper L-weighting of G. •

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Let

Amax = VL,max( G) n A, Bmax = VL,max(G) n B,

Amin = VL,min (G) n A, Bmin = VL ,min(G) n B. By Claim 2, none of the sets Amax, Bmax, Amin, Bmin is empty.

Claim 3. IA max U Amini = IBmax U Bminl = 1. Proof. Assume to the contrary that IA max U Amini> 1. Then there exist v E Amax and Vi E A min such that v =I Vi . Let fo be the partial L-weighting of G that max-weights v, and let L 1 be the (1,2)-total weighting of G1 = G - v induced by Land fo. Then for any vertex x E B ,

>

=

Therefore Vi E VL1,min(G 1) and VL1,min(G1) n B =

0.

Let 81 = 8 0 U QL,max(V) and T 1 = To - Nc[v]. We have T1 = 0 and 8 1 = B max · Let II be the partial L 1-weighting of G 1 that min-weights Vi and let L2 be the (1,2)-total list assignment of G2 = G1 - Vi induced by L 1 and II· Let 82 = 8 1 - N C1 [Vi] and T 2 = T1 U Q £1 ,min (Vi). It is obvious that 82 = 0. Since VL1,min(G1) n B = 0, we have T2 = QL1,min(V' ) = 0. By the minimality of G, we know that G2 has a proper (L2 ; 82, T2)-weighting. By Theorem 2.3, G1 has a proper (L 1 ; 8 1 , T1)-weighting and G has a proper (L; 8 0, To)-weighting. _ Assume Amax = A min = {v} and Bmax = Bmin = {u} . Let fo be the partial L-weighting of G that max-weights v, and let L 1 be the (1, 2)-totallist assignment of G 1 = G - V induced by Land fo. Then 8 1 = 80 U QL,max(V) = {u} and T1 = To - Nc[v] = 0. By Theorem 2.3, if G1 has a proper (L 1 ; 8 1 , T1)-weighting, then G has a proper (L; 80, To)weighting. As 8 1 =10, we cannot conclude (from the minimality of G) that G 1 has a proper (L 1 ; 8 1, TI}-weighting. If VL1 ,min(GI) ~ A , then let Vi E VLI,min(GI} . Let II be the partial L 1weighting that min-weights Vi. Let L 2 be the (1,2)-total list assignment

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T.-L. Wong, D. Yang and X. Zhu

of G 2

= G1

-

v' induced by L1 and h. Let 8 2 = 8 1

-

Ne[v'] and

T 1 U QLI,min(V'). Then 8 2 = T2 = 0. Since G2 =1= K 2 , by the minimality of G, G2 has a proper (L2 ; 8 2, T2)-weighting. By Theorem 2.3, this implies that G 1 has a proper (L 1 ; 8 1 , Td-weighting, and hence G has a proper (L; 80, To)-weighting. Thus we may assume that VLI,min(G 1 ) n B =1= 0. Let

T2

=

B' = VLI,min(Gd n B. By definition, for each x E B', for each yEA - {v},

Claim 4. For each x E B' and for each yEA - {v},
<
Interchange the roles of A and B, and let f~ be the partial L-weighting of G that max-weights u, let L~ be the (1, 2)-weighting of G~ = G - u induced by L and f~, and let A' = VL~,min(GD n A. By symmetry, we have A'

=1=

0 and the following claim holds.

Claim 5. For each yEA' and for each x E B - {u} ,
<
By combining Claims 4 and 5, we can prove the following claim.

Claim 6. A' = {v} and B' = {u} . Proof. If A' - {v} =1= 0 and B' - {u} =1= 0, then for yEA' - {v} and x E B' - {u}, Claims 4 and 5 contradicts each other. Thus, by symmetry, we may assume that B' - {u} = 0 and hence B' = {u} (as B' =1= 0). Assume the claim is not true. Then A' - {v} =1= 0. Let sEA' - {v}. By definition,

As
+ Lmax(e) -

Lmin(e)

=
=
Hence

=
But B' = {u} implies that
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List Total Weighting of Graphs

Let w i= u, v be a vertex such that for any x i= u, v, IPL,max (x) ::; IPL,max(W) . Without loss of generality, we assume that wEB. Then in the graph G 1 = G - v, W is a local maximum vertex with respect to L 1 . Let h be the partial L 1-weighting of G 1 that max-weights w. Let L 2 be the (1,2)-total list assignment of G2 = G1 - w induced by L 1 and h. Let 82 = 8 1 U QLl,max(w) and T2 = T 1 - NCI [w] . Then T2 = 0 and 8 2 ~ {u} U (A - {v}) . Now we shall prove that VL2,min(G2) = {u}. I.e., for any z E (A-{v}) U

(B - {u,w}) , If z E B - {u, w}, then since B' = {u} , it follows that

If Z E A - {v}, then

Let h be the partial L 2-weighting of G 2 that min-weights u, and let La be the (1,2)-total list assignment of Ga = G2 - u induced by L 2 and h. Let 8 a = 8 2 - NC2[U] and Ta = T2 U Q L2 ,min(U) . Then 8a = Ta = 0. Since i= K2· Hence has a proper (La ;s; Ta)-weighting. G i= K2,a, hence By repeatedly applying Theorem 2.3, we conclude that G 2 has a proper (L 2 ; 8 2, T2)-weighting , and hence G 1 has a proper (L 1 j 8 1, T1)-weighting, and hence G has a proper (L j 8 0 , To)-weighting. _

c,

c,

It was shown in [7] that complete bipartite graphs other than K2 are 3-edge weight choosable. By assigning to every vertex weight 0, we have the following corollary. Corollary 3.3. Every complete bipartite graphs other than K 2 are 2-edge weight choosable.

Corollary 3.3 can be stated as a result about matrices, which seems interesting. Corollary 3.4. If n+m ~ 3, and E = [L i j ] is an n x m matrix where each L i j is a set of two real numbers, then there is an n x m real matrix A = [aij] such that aij E L i j and no row sum of A is equal to a column sum of A .

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T.-L . Wong, D. Yang and X. Zhu

Note that the induction proof of Theorem 3.2 cannot be used to give a direct proof of Corollary 3.3. This shows that even if our interests are in edge weight choosability of graphs, it can be helpful by considering the more general total weight choosability of graphs.

4. ALGEBRAIC METHOD Assume G = (V, E) is a simple graph, where V = {VI, V2, , v n } and E = {el, e2, ... , em}. Denote by X the set of variables XVI' X V2, , X Vn' X el , X e2, ... , x em' For each vertex v of G, let Qv = I:eEE(v) X e + XV' Fix an arbitrary orientation

Gof G.

Let

p(G)=P(XVllXV2 "",XVn,XellXe2"",Xem)=

IT

(Qv-Qu).

UVEE(G)

For each edge uv (oriented from u to v), QV - Qu is a polynomial of degree 1 with variable set X. So p(G) is a polynomial of degree m = IE(G)I with variable set X. For different orientations G of G, the polynomials defined may differ by a sign, which is irrelevant for our purpose. Suppose f is a total weighting of G. By definition, for each vertex u,

is equal to the evaluation of the polynomial Qu with Xv = f(v) for each v E V and X e = f (e) for each edge e E E. Therefore f is a proper total weighting of G if and only if

A mapping TJ : VUE -+ {O, 1, ... } is called a valid index function if I:yEvuE TJ(Y) = m . For a valid index function TJ, let Cq be the coefficient of the monomial ITyEvuE x~(y) in the expansion of p( G). It follows from the Combinatorial Nullstellensatz [6,5] that if Cry =/= 0, and IL(y)1 ~ TJ(Y) + 1 for all y E VUE , then there exist f(y) E L(y) for y E VuE such that

I.e., G has a proper L-total weighting.

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List Total Weighting of Graphs

The coefficient Cry of the monomial ITyEvuE x~(y) is associated to the permanent of some matrices. Let A c be the m x (n + m) matrix, with

rows indexed by the edges of G and columns indexed by vertices and edges, defined as follows: If e = uv is an edge of G oriented from u to v , then let

aey =

{

= v or y i= e is an edge incident to v

1,

if y

-1,

if y = u or y

0,

otherwise.

i= e is an edge incident to

u

Then

So

For each y E VUE , we denote by A y the column of A c indexed by y.

A c = [AV1 ' AV2 "

' "

Av n , Ae l , Ae2 , · · · , Ae m ] ·

For a valid index function 1], let A c(1]) be the m x m matrix in which the column A y occurs 1](y) times for each y E VUE. It is known [4] that Cry i= 0 if and only if the permanent of A c(1]) is non-zero, where the permanent of an m x m matrix A = [aij] is defined as m

per (A) =

L II a

aja(j ),

j=l

here the summation is taken over all the permutations a of {1, 2, ... ,m}. Thus by showing that certain matrices have non-zero permanent, we can prove results about total weight choosability of graphs. For any graph G = (V, E), let 1]0 be the valid index function defined as 1]o(e) = 1 for each edge e and 1]o(v) = 0 for each vertex v. By the discussion above, if A c (1]o ) has non-zero permanent, then G is (1,2)-choosable. It is not difficult to verify that for G = K 1,2 , K 1,3 , K 2,2, K2,3 , the permanent of A c (1]o ) is nonzero. Therefore these graphs are (1, 2)-choosable. We believe that Conjectures 1.1 and 1.2 can be proved by using this method. To be precise, the following stronger conjectures are proposed in [7,13]: Conjecture 4.1 [13]. For any graph G, there is a valid index function with TJ{Y) ~ 1 for all Y E VUE such that per (A 6 (TJ») i= O.

1]

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T.-L. Wong, D. Yang and X. Zhu

Conjecture 4.2 [7]. For any graph G without isolated edges, there is a valid index function TJ with TJ( v) = 0 for all v E V and TJ( e) ::; 2 for all e E E such that per ( A e(TJ)) =!= O.

In [13], we had limited success and proved that if G = K 2 ,n, then G is (1,2)-choosable, and if G is a complete graph, or G = K3,n, or G is a tree or a generalized theta graph, then Conjecture 4.1 holds for G, and hence G is (2,2)-choosable. We were unable to prove that general complete bipartite graphs are (2,2)-choosable by this method in [13]. In this paper, by using the max-min weighting method, we are able to prove a stronger result: if G is a complete bipartite graph, then G is (1,2)-choosable. The main result in [13] that complete graphs are (2,2)-choosable is also implied by Theorem 3.1. In this sense, the max-min method has been more successful. However, it seems to us that the hope of confirming Conjectures 1.1 and 1.2 is more likely to lie on the algebraic method. For this purpose, we would still like to see a proof of the results in this paper by the algebraic method. Because complete bipartite graphs other than K 2 are (1, 2)-choosability, one naturally asks whether per (Ae(TJc)) =!= 0 for all complete bipartite graphs G =!= K2. But this question has a negative answer. Theorem 4.3. If G is a bipartite graph with an odd number of edges and there is an automorphism of G that interchanges the two partite sets, then per (Aq(TJe)) = O. In particular, if n is odd and G = Kn,n, then per ( Ae( TJe)) = O. Proof. The permanent of the matrix Ae(TJe) depends on the orientation of the edges. Changing the orientation of one edge will change the sign of the permanent. Assume the two partite sets of G are A and B . Let 6 1 be the orientation of G in such a way that all the edges are from B to A. Let 6 2 be the orientation of G in which all the edges are oriented from A to B. Since 62 is obtained from 61 by changing the orientation of an odd number of edges, their permanents differ by a sign. However, by our assumption, 62 and 61 are isomorphic, so they have the same permaments. Therefore their common permanent is O. •

This result shows that not every result on total weight choosability of graphs can be proved by the method of calculating the permanents of some matrices. Nevertheless, it has not yet crushed our faith on Conjectures 4.1 and 4.2. For complete bipartite graphs, based on some experiments, we suspect that if n =!= m or n = m are even, then for G = Kn,m, Ae(TJc) has nonzero permanent. We can prove this for some special cases.

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List Total Weighting of Graphs

Theorem 4.4. If n, q are positive integers such that n has a prime factor p> q, then for G = Kn,q, A6(17c) has nonzero permanent. Proof. As the implication of this theorem to total choosability of graphs is already proved, we shall just sketch an outline of the proof and omit some details. Assume the two partite sets of G are A = {VI, V2, ... , vn } and B = { U 1, U2, . .. , u q } . We orient the edges of G so that all the edges are oriented from B to A. We shall prove that per ( A6( 17c)) is not a multiple of p , and hence per (A6(17c)) # o. Suppose e = uv is an edge of G oriented from U to V , then the column Ae can be expressed as Ae = Av + Au. It is known (and easy to see) that the following is true: Assume A is a matrix, and Ai is a column of A, and Ai = A~ + A~' . Let A' be obtained from A by replacing A with A~ and A" be obtained from A by replacing Ai by A~'. Then per (A) = per (A')

(1)

+ per (A") .

The following observation can be derived from the definition of permanent

(cf. [4]) Observation 4.5. If M is an integer matrix, and k of the columns of M are identical, then per (M) is a multiple of kL Suppose F is a subset of E and F' is a subset of F. Let AF',F be the matrix obtained from A6( 17 by replacing, for each e = uv E F' , the column A e by Au , and replacing for each e = uv E F - F', the column A e by A v . For any subset F of E, by repeatedly applying (1), we have

c)

per (A 6(17c)) = Let FI =

{UIVj :

j

L

per (AF',F).

F'r;F

= 1,2, . . . , n} . For k = 0,1 , ... , n, let FI ,k =

{ F' ~ H : IF'I = k} .

It is easy to see that if F', F" E FI ,k, then

If IF'I = k and k ~ p, then by Observation 4.5, multiple of k! , whi ch is a multiple of p .

!per (AF',Fl) I is

a

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T.-L. Wong, D. Yang and X. Zhu

If 1 :s; k < p, then IF1,kI = (~) is a multiple of p. Hence the sum .EF'EFl,k per (AF',Fl) is a multiple of p. Therefore

Let F2

= {UjV s

: S

= 1,2, ... , n,

per (A0,Fl) =

j

= 1, 2}.

L

Then

per (AF',F2)'

F'c;,F2, F'nFl =0 Repeat the above process, we can prove that

For i ~ 2, let F; = {UjV s : S = 1,2, .. . ,n, j the process above, we can prove that

= 1,2, .. . , i}. By repeating

Therefore per (AeC'7c)) ~ per (A0,E) But it is easy to verify that per (A0,E)

= (q!t

(mod p). ~ 0 (mod p).

•

It would be interesting to prove that for any complete bipartite graph G = Kn ,m, unless n = m is odd, we have per( A G(7Jc )) =1= O. If this is true, then it follows from a result in [13] that Conjecture 4.1 holds for all complete bipartite graphs.

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L. Addario-Berry, R. E. L. Aldred, K. Dalal and B. A. Reed , Vertex colouring edge partitions, J. Gombin. Th eory Ser. B, 94 (2005), 237-244.

[2]

L. Addario-Berry, K. Dalal and B. A. Reed, Degree constrianed subgraphs, Proceedings of GRAC02005 , Volume 19, Electron. Notes Discrete Math., Amsterdam (2005), 257-263, Elsevier.

[3]

L. Addario-Berry, K. Dalal , C. McDiarmid, B. A. Reed and A. Thomason, Vertexcolouring edge-weight ings, Combinatorica, 27 (2007), 1-12.

[4J

N. Alon and M. Tarsi, A nowhere zero point in linear mappings, Gombinatorica, 9 (1989), 393-395 .

353

List Total Weighting of Graphs [5]

N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica, 12 (1992), 125-134.

[6]

N. Alon and M. Tarsi, Combinatorial Nullstellensatz, Combin. Prob. Comput ., 8 (1999), 7-29.

[7]

T . Bartnicki, J . Grytczuk and S. Niwczyk, Weight choosability of graphs, Journal of Graph Theory, 60(3) (2009), 242-256.

[8]

G. Chang, C. Lu , J . Wu and Q. Yu, Vertex coloring 2-edge weighting of bipartite graphs, preprint, 2007.

[9]

M. Karoriski, T . Luczak and A. Thomason, Edge weights and vertex colour, J. Combin. Theory Ser . B, 91 (2004), 151-157.

[10]

M. Kalkowski, M. Karoriski and F. Pfender, Vertex-coloring edge-weightings: towards the 1-2-3-Conjecture, Journal of Combinatorial Theory, Ser . B, to appear.

[11]

J . Przybylo and M. Wozniak, 1,2 -conjecture, Preprint MD 024, www.ii.uj.edu.pl/preMD/

[12]

J . Przybylo and M. Wozniak, 1,2-conjecture II, Preprint MD 026, www.ii.uj .edu .pl/preMD/

[13]

T. Wong and X. Zhu, Total weight choosability of graphs, Journal of Graph Theory, to appear.

Tsai-Lien Wong

Daqing Yang

Department of Applied Mathematics National Sun Vat-sen University

Center for Discrete Mathematics Fuzhou University Fuzhou, Fujian 350002

e-mail :

tlYong~ath .nsysu.edu .tY

China e-mail:

Xuding Zhu Department of Applied Mathematics National Sun Vat-sen University Kaohsiung, Taiwan 80424 and National Center for Theoretical Sciences e-mail :

zhu~ath.nsysu .edu.tY

daqing85~yahoo.com

Fete of Combinatorics, pp . 355-365 .

BOlYAI SOCIETY MATHEMATICAL STUDIES, 20

OPEN PROBLEMS

1.

THE

GROTHENDIECK

CONSTANT AND THE

Lov ASZ

THETA

FUNCTION (AN OPEN PROBLEM)

Noga Alon (Department of Mathematics and Computer Science, Tel Aviv University, e-mail: [email protected]) PROBLEM POSER:

The Grothendieck constant K(G) of a graph G constant K such that for every A : E -+ R , sup

L

= (V, E) is the least

A(u,v) . (f(u),f(v))

j :v-+SIVI-l {u,V}EE

~K

sup

L

A(u,v) . f(u)f(v) .

j :V-+{-l ,+l} {u,V}EE

This notion was introduced in [2], where it is shown that there is an absolute positive constant c so that for every graph G, K(G) ~ clog 'l9(G) , with 'l9(G) being the Lovasz theta function of the complement of G, defined in [12]. Problem. Is there a lower bound for K(G) in terms of'l9(G)? In particular, is there an absolute constant c' > 0 so that for every graph G, K(G) :2: c'log'l9(G)? The results in [2] imply that this is true for perfect graphs (and in fact for any family of graphs in which the chromatic number and the maximum clique are polynomially related), and the results in [1] show that this is true for random graphs and for certain pseudo-random graphs .

356 2.

Open Problems

QUADRUPARTIONS

PROBLEM POSER: Imre Barany (Alfred Renyi Institute of Mathematics, Budapest, Hungary, e-mail: barany@renyLhu) CO-AUTHOR: Branko Griinbaum (Department of Mathematics, University of Washington) A few decades ago, Branko Griinbaum raised the following problem which has never appeared in print: Conjecture 1. If K is a convex set in the plane whose area is one, then for each t E [0,1/4] there exist two orthogonal lines that partition K in four pieces of areas t , t , (1/2 - t), (1/2 - t) in clockwise order. An elementary continuity (or topological) argument shows that the conjecture is true when t = and when t = 1/4. It seems that topology, or topology alone, is not enough for the general case.

°

Here is an even stronger version of the conjecture. Assume that a nice probability measure is given in the plane. (Here "nice" means, say, that it is a Borel measure which is zero on every line segment.) Again, let t E [0,1/4] . Is it true, then, that there exists a pair of orthogonal lines partitioning the plane into four pieces with measures t, t, 1/2 - t , 1/2 - t in clockwise order? (Again, this is true for t = and t = 1/4.) We venture to conjecture that the answer in general is "no":

°

Conjecture 2. There is a nice probability measure in the plane, so that, for some t E (O, 1/4), no pair of orthogonal lines partitions the plane into four pieces of measure t , t, 1/2 - t, 1/2 - t in clockwise order.

3.

CHROMATIC NUMBER OF A RANDOM SUBGRAPH

PROBLEM POSER: Boris Bukh (Princeton University / UCLA, e-mail: B.Bukh@dpmms .cam.uc.uk) Chromatic number of a graph G is denoted X{G). Let Gp be the random subgraph of G obtained from G by keeping each edge with probability p and removing it with probability 1 - p independently at random. For example,

357

Open Problems

G 1/ 2 is the subgraph chosen uniformly at random among all subgraphs of G. It is easy to see that whenever lip is a positive integer. Is there an

€

> 0 such

that

Perhaps it is even true that

X(G)

p ) ~ c(P)logx(G) IE [ X(G]

for every fixed 0 < p < 1, and a positive constant c(p) depending only on p. It is known that the inequality

holds.

4. Two CONJECTURES ON QUASI-KERNELS PROBLEM POSER: Peter L. Erdos (Alfred Renyi Institute of Mathematics, Budapest, Hungary, e-mail: elp@renyLhu) Let D = (V, E) be (finite or infinite) directed graph. An independent vertex subset A c V is a quasi-kernel (also known as semi-kernel) iff for each point v there is a path of length at most 2 from some point of A to v . Similarly, the independent vertex subset B c V is a quasi-sink, iff for each point v there is a path of length at most 2 from v to some point of A . It is a well-known fact, that every finite (table-tennis) tournament has a (single-point) quasi-kernel (quasi-sink) . In 1973 the following nice generalization was proved: Theorem 1 (V. Chvatal - L. Lovasz [5]). contains a quasi-kernel (quasi-sink).

Every finite directed graph

In 1976 the following conjecture was stated:

358

Open Problems

Conjecture 2 (P. 1. Erdos - L. A. Szekely). Assume that in the finite digraph D = (V, E) for each vertex v E V the indegree d- (v) ;::: 1. Then there exists a quasi-kernel A with the property IAI :::; 1V1/2. For example the disjoint union of oriented 0 4 's satisfies this with equality. Theorem 1 does not hold in case of infinite directed graphs: for example if Z denotes the directed graph of all integers where each edge is directed "upwards" , clearly there is nor quasi-kernel neither quasi-sink. However its vertex set can be easily partitioned into two subsets, such that one spanned sub-tournament contains a quasi-kernel, while the other one contains a quasi-sink.

Conjecture 3 (P. 1. Erdos - L. Soukup (2008) [8]). Every (countable) infinite directed graph D can be partitioned into two vertex classes, such that one spanned subgraph contains a quasi-kernel, while the other one contains a quasi-sink.

5.

PARTITIONING GRAPHS INTO LARGE STARS

PROBLEM POSER: Jan Kratochvil (Department of Applied Mathematics, Charles University, Prague, Czech Republic, e-mail: [email protected]) Informally, the question is to identify sufficient conditions that would guarantee that the vertex set of a graph under consideration can be partitioned into large enough stars. In particular, we ask whether large minimum degree or large vertex connectedness are such sufficient conditions. Formally, for a graph G = (V, E), we define sc (G) to be the maximum s such that G can be partitioned into disjoint stars with at least s rays each (i.e. V = U~=l Vi , such that for each i, IViI ;::: s + 1 and G[Vi] contains a vertex adjacent to all other its vertices) . We further define two functions

sco(k) = scK(k) =

min

G :o(G)2k

sc (G),

min

G k-vertex-connected

Problems. 1) Is limk-+oosco(k) = oo?

and sc(G) .

359

Open Problems

2) Is limk-too sc, (k) = oo? 3) If yes, determine, as tight as possible, the growth rates of these functions.

Comments. The motivation for the problem comes from TCS , namely exact exponential time algorithms for the so called £(2, l l-labeling problem (joint work with F. Havet , M. Klazar, D. Kratsch, and M. Liedloff). The problem has been solved by N. Alon and N. Wormald; see their article in the present volume.

6.

OPEN PROBLEMS ON GIRTH

PROBLEM POSER: Nathan Linial (School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel , e-mail: [email protected]) We recall that the girth of a graph G is the length of the shortest cycle in G. Let us denote by g(d, n) the highest girth of ad-regular n-vertex graph. Our main interest is is the case where d ~ 3 is fixed and n is large. The current best asymptotic bounds on this function are logn (2+0(1))log(d_l) ~g(d,n)~

(4) 3-

logn 0(1) log(d-l)'

Both sides of this inequality raise interesting questions. The lower bound comes from a famous paper by Lubotzky Phillips and Sarnak and is based on deep results from representation theory. Using combinatorial and probabilistic methods it is easy to prove the lower bound with coefficient Any new proof using combinatorial and probabilistic proof 1 instead of that yields 1 + EO for some EO that is bounded away from zero would be of interest.

1.

The upper bound (also called the Moore Bound) is quite obvious. I conjecture, though, that there is some 8 > 0 such that for every integer d ~ 3 and large enough n, there holds: logn (2 - 8) log (d _ 1) ~ g(d,n).

360 7.

Open Problems

CHROMATIC NUMBER OF SUBGRAPH

PROBLEM POSER: Laszlo Lovasz (Eotvos Lorand University, Budapest, Hungary, e-mail : [email protected]) An old problem: Is it true that every simple graph without isolated nodes that is not complete contains two adjacent nodes u and v such that X(G - u - v) ~ X(G) -1?

8.

EDMONDS GRAPHS

PROBLEM POSER: Claudio L. Lucchesi (University of Campinas, Sao Paulo, Brazil, e-mail: [email protected]) CO-AUTHORS: Marcelo H. de Carvalho (UFMS - Brazil), U. S. R. Murty (University of Waterloo - Canada) Edmonds 1965. A vector x in RE belongs to the perfect matching polytope of a graph G iff:

x(e)

~

0 'lie E E

x(8(v)) = 1 VvEV

x( 8(S))

~ 1

vs c

V, lSI odd

(non-negativity) (degree constraints) (odd set constraints)

Bipartite graphs do not need odd set constraints, An Edmonds graph is one for which the odd set constraints are needed. Question. Is the class of Edmonds graphs in co-NP? We know that the class is in NP . Let G be a graph. Graph G is matching covered if it is non-trivial, connected and every edge lies in a perfect matching. For a subset S of V , the cut 8(S) consists of those edges having precisely one end in S . A cut 8(S) is tight if every perfect matching of G has precisely one edge in 8(S) , and it is separating if both G/ Sand G /8 are also matching covered. Every tight cut is separating, but not every separating cut is tight. In a bipartite graph, every separating cut is tight. Carvalho, Lucchesi, Murty (JCT-B, 2004). A brick G is Edmonds iff it has a separating cut that is not tight.

Open Problems

361

Reed, Wakabayashi 2004. A brick is Edmonds iff it has two disjoint odd cycles 01 and 02 such that G - V(Ol) - V(02) has a perfect matching. Corollary. If a graph is Edmonds then it must have two disjoint odd cycles. Example. Odd wheels and Mobius ladders are not Edmonds. We have shown [Discrete Math., 2006] that, essentially, the only planar matching covered graphs that are not Edmonds are the bipartite graphs and the odd wheels.

9.

COUNTING CONVEX COLORINGS OF GRAPHS

Johann A. Makowsky (Dept . of Computer Science, Technion Haifa, Israel, e-mail: [email protected]) PROBLEM POSER:

A vertex coloring of a graph G = (V, E) with k colors (k E N) is a function I : V ~ [k] . I is convex if for every i E [k] the colorclass 1- 1 (i ) induces a connected subgraph. For a partial function 10 : V ~ [k] we say that is convex if there a is a total function I extending 10 which is convex. In this case we also say that I is a convex extension of 10. Convex extensions of partial colorings of trees have been introduced in the context of phylogenetic trees by S. Moran and S. Snir [15]. The existence problem of convex colorings for an arbitrary graph G is easily solved by trying to color every connected component by one color, and only depends on the number of colors available and the number of connected components of G. It follows from [13, 10] that the number of convex colorings of a graph G is a polynomial in k, which we denote by conv (G, k). For k = 1 we have conv (G, 1) = 1, if G is connected, and conv (G, 1) = 0 otherwise.

io

Problem. What is the complexity of computing conv (G, 2)? We think that this problem is ~P-complete. Let En be the empty graph on n vertices. One easily verifies that conv (G u En, k) = conv (G, k - n) . (k . (k - 1) . . . . . (k - n + 1)) where U denotes the disjoint union of graphs . This shows, similar as in [11], that, if evaluating conv (G, 2) is ~P-hard , so is evaluating conv (G, k) for k? 2.

362

Open Problems

Using methods from [7] one can prove that for fixed k, evaluating conv (G, k) is P-time computable on graphs of tree-width at most t.

Comments. The problem has been solved by A. J. Goodall an S. D. Noble, see [9].

10.

MAXIMUM NUMBER OF K r,r SUBGRAPHS

Daniel Marx (Budapest University of Technology and Economics, Budapest, Hungary, e-mail: [email protected]) PROBLEM POSER:

Let G be an n-vertex graph that has no K r ,r+1 subgraph. What is the maximum number of Kr,r subgraphs in the graph? In particular, can we give an upper bound of the form O(n C ) (for some absolute constant c) or of the form f(r) . n C (for some function f and absolute constant c)? Is it possible to construct graphs where this number is n f2(r )?

11.

ON HAJNAL'S TRIANGLE-FREE GAME

Akos Seress (The Ohio State University, e-mail: [email protected]) PROBLEM POSER:

Andras Hajnal proposed the following graph game in the early 1990's. Beginning with the empty graph on n vertices, two players, A and B, alternatingly draw edges. We do not distinguish the edges drawn by A and B. The loser is the player who cannot make a move without completing a triangle. The winner of the game is known only for n :::; 15 [3, 17, 19] and the main problem is to determine the winner for all n: Some results were obtained in variants of the game, when we place restrictions on the graph created by the players. If the non-isolated vertices of the graph drawn must always be in one connected component then the winner depends only on the parity of n [19]. In another variant, we require that the graph created by the players has maximal degree at most k. This version is surprisingly difficult even for k = 3, and the winner is known only for k = 2 and k = 3 [14]. After some erratic behaviour for small n , the

363

Open Problems

winner depends on the value n mod 2 and n mod 4 in the cases k = 2 and 3, respectively. The restricted degree variant makes sense even if we drop the triangle-free requirement, and also in this version the winner is known only for k = 2 and k = 3 [14] .

12. Is Two

COLORING ALWAYS POLYNOMIAL?

PROBLEM POSER: Joel Spencer (Department of Computer Science and Department of Mathematics, Courant Institute, New York, USA, e-mail: [email protected]) Suppose A a is a family of k-element subsets of an n-set n with each set intersecting at most d other sets. The Lovasz Local Lemma says that if 4d21- k < 1 (this can be slightly strengthened, for the best bound, see [18]) there necessarily exists a two coloring of n with no A a monochromatic. Is there, for every such fixed d, k a polynomial (in n) time algorithm that finds that coloring? Beck has shown that the answer is yes for a wide class of d, k. We conjecture the general answer is no. In somewhat imprecise complexity terms : that there are fixed d, k for which the problem of finding a coloring can be placed in a class of problems which are generally regarded as not solvable in polynomial time .

Added in Proof. Such a randomized polynomial time algorithm has been found recently, see [16J . The existence of a deterministic polynomial time algorithm is still open in this generality, but such an algorithm is given in case the slightly stronger condition d = O(2(1-e)k) holds, see [4] .

13. PIZZA PROBLEM PROBLEM POSER: Peter Winkler (Department of Mathematics, Dartmouth, Hanover, New Hampshire, USA, e-mail: [email protected]) Alice and Bob share a pizza; the pizza is round and sliced by radial cuts into some arbitrary number of pieces of various sizes. Alice chooses any slice to eat , after which Bob and Alice alternate taking slices, but must always take a slice which is adjacent to some previously-taken slice.

364

Open Problems

Conjecture: Alice can always get at least 4/9 of the pizza. Note: 4/9 is best possible; 1/3 is easy. Comment: based on a puzzle by Dan Brown which asked whether Alice can guarantee 1/2.

Comments. The problem has been solved by J. Cibulka, J. Kyncl, v. Meszaros, R. Stolar and P. Valtr; see their article in the present volume. The problem also has been solved simultaneously and independently by the group consisting of K. Knaurer, T. Uekerdt and P. Micek.

REFERENCES [1]

N. Alon and E. Berger, The Grothendieck constant of random and pseudo-random graphs, Discrete Optim., 5 (2008), no. 2, 323-327.

[2]

N. Alon, K. Makarychev, Y. Makarychev and A. Naor, Quadratic forms on graphs, Invent. Math., 163 (2006), no. 3, 499-522 .

[3]

S. C. Cater, F . Harary and R. W. Robinson, One-color triangle avoidance games, Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001), vol. 153,2001 , pp. 211-22l.

[4] K. Chandrasekaran, N. Goyal and B. Haeupler, Det ermininistic algorithms for the Lovasz Local Lemma , in: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithm (SODA 2010). [5]

V. Chvatal and L. Lovasz, Every directed graph has a semi-kernel, Hypergraph Seminar (Proc. First Working Sem., Ohio State Univ., Columbus , Ohio, 1972; dedicated to Arnold Ross), Springer, Berlin , 1974, pp. 175. Lecture Notes in Math., Vol. 41l.

[6]

J . Cibulka, J . Kyncl, V. Meszaros, R. Stolar and P. Valtr, Solution of Pet er Winkler 's pizza problem, this volume.

[7]

B. Courcelle, J. A. Makowsky and U. Rotics , On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic, Discrete Appl. Math ., 108 (2001), no. 1-2, 23-52, International Workshop on GraphTheoretic Concepts in Computer Science (Smolenice Castle, 1998).

[8]

P. L. Erdos and L. Soukup, Quasi-kernels and quasi-sinks in infinite graphs, Discrete Mathematics, 309 (2009), no. 10, 3040-3048.

[9]

A. J. Goodall and S. D. Noble, Counting cocircuits and convex two-colourings is #P-complete, preprint, arXiv : 0810.2042, 2008. http://arxi v . org/pdf /0810.2042

[10]

T . Kotek, J. A. Makowsky and B. Zilber, On counting generalized colorings, CSL '08: Proceedings of the 22nd international workshop on Computer Science Logic (Berlin , Heidelberg), Springer-Verlag, 2008, pp. 339-353.

365

Open Problems [11]

N. Linial, Hard enumeration problems in geometry and combinatorics, SIAM J. Algebraic Discrete Methods, 7 (1986), no. 2, 331-335.

[12] L. Lovasz, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), no. 1, 1-7. [13]

J. A. Makowsky and B. Zilber, Polynomial invariants of graphs and totally categorical theories, MODNET Preprint No. 21, http://www .logique.jussieu .fr/modnet/Publications/Preprint%20 server, 2006.

[14]

N. Mehta and

A. Seress,

Graph Games, manuscript, in preparation.

[15] S. Moran and S. Snir, Efficient approximation of convex recolorings, J. Comput. System Sci., 73 (2007), no. 7, 1078-1089 . [16] R. Moser and G. Tardos , A constructive proof of the general Lovasz Local Lemma, Journal of the ACM, 57 (2010) (2), Art. 11. [17] P. Pralat, A note on the one-colour avoidance game on graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 9pp (programs in C/C++) , to appear. [18] J. Radhakrishnan and A. Srinivasan, Improved bounds and algorithms for hypergraps 2-coloring, Random Structures Algorithms, 16(3) (2000). [19]

A. Seress, On Hajnal's triangle-free game,

Graphs Combin., 8 (1992), no. 1,75-79.